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## G = C7×C22.56C24order 448 = 26·7

### Direct product of C7 and C22.56C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C22.56C24
 Chief series C1 — C2 — C22 — C2×C14 — C22×C14 — D4×C14 — C7×C4.4D4 — C7×C22.56C24
 Lower central C1 — C22 — C7×C22.56C24
 Upper central C1 — C2×C14 — C7×C22.56C24

Generators and relations for C7×C22.56C24
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=g2=1, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >

Subgroups: 362 in 220 conjugacy classes, 142 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2×C14, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22.56C24, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C4⋊D4, C7×C22⋊Q8, C7×C22.D4, C7×C4.4D4, C7×C42.C2, C7×C22.56C24
Quotients: C1, C2, C22, C7, C23, C14, C24, C2×C14, 2+ 1+4, 2- 1+4, C22×C14, C22.56C24, C23×C14, C7×2+ 1+4, C7×2- 1+4, C7×C22.56C24

Smallest permutation representation of C7×C22.56C24
On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 222)(9 223)(10 224)(11 218)(12 219)(13 220)(14 221)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 68)(58 69)(59 70)(60 64)(61 65)(62 66)(63 67)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 96)(86 97)(87 98)(88 92)(89 93)(90 94)(91 95)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 124)(114 125)(115 126)(116 120)(117 121)(118 122)(119 123)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 152)(142 153)(143 154)(144 148)(145 149)(146 150)(147 151)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 180)(170 181)(171 182)(172 176)(173 177)(174 178)(175 179)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 208)(198 209)(199 210)(200 204)(201 205)(202 206)(203 207)
(1 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 41)(30 42)(31 36)(32 37)(33 38)(34 39)(35 40)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)(85 103)(86 104)(87 105)(88 99)(89 100)(90 101)(91 102)(106 123)(107 124)(108 125)(109 126)(110 120)(111 121)(112 122)(113 131)(114 132)(115 133)(116 127)(117 128)(118 129)(119 130)(134 151)(135 152)(136 153)(137 154)(138 148)(139 149)(140 150)(141 159)(142 160)(143 161)(144 155)(145 156)(146 157)(147 158)(162 179)(163 180)(164 181)(165 182)(166 176)(167 177)(168 178)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(190 207)(191 208)(192 209)(193 210)(194 204)(195 205)(196 206)(197 215)(198 216)(199 217)(200 211)(201 212)(202 213)(203 214)
(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 117)(16 118)(17 119)(18 113)(19 114)(20 115)(21 116)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 120)(29 136)(30 137)(31 138)(32 139)(33 140)(34 134)(35 135)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 144)(44 145)(45 146)(46 147)(47 141)(48 142)(49 143)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(8 28)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 223)(16 224)(17 218)(18 219)(19 220)(20 221)(21 222)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 68)(58 69)(59 70)(60 64)(61 65)(62 66)(63 67)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(106 123)(107 124)(108 125)(109 126)(110 120)(111 121)(112 122)(113 131)(114 132)(115 133)(116 127)(117 128)(118 129)(119 130)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 152)(142 153)(143 154)(144 148)(145 149)(146 150)(147 151)(190 207)(191 208)(192 209)(193 210)(194 204)(195 205)(196 206)(197 215)(198 216)(199 217)(200 211)(201 212)(202 213)(203 214)
(1 78 34 102)(2 79 35 103)(3 80 29 104)(4 81 30 105)(5 82 31 99)(6 83 32 100)(7 84 33 101)(8 172 222 176)(9 173 223 177)(10 174 224 178)(11 175 218 179)(12 169 219 180)(13 170 220 181)(14 171 221 182)(15 184 22 167)(16 185 23 168)(17 186 24 162)(18 187 25 163)(19 188 26 164)(20 189 27 165)(21 183 28 166)(36 88 43 92)(37 89 44 93)(38 90 45 94)(39 91 46 95)(40 85 47 96)(41 86 48 97)(42 87 49 98)(50 130 74 106)(51 131 75 107)(52 132 76 108)(53 133 77 109)(54 127 71 110)(55 128 72 111)(56 129 73 112)(57 124 68 113)(58 125 69 114)(59 126 70 115)(60 120 64 116)(61 121 65 117)(62 122 66 118)(63 123 67 119)(134 203 158 207)(135 197 159 208)(136 198 160 209)(137 199 161 210)(138 200 155 204)(139 201 156 205)(140 202 157 206)(141 191 152 215)(142 192 153 216)(143 193 154 217)(144 194 148 211)(145 195 149 212)(146 196 150 213)(147 190 151 214)
(1 74)(2 75)(3 76)(4 77)(5 71)(6 72)(7 73)(8 204)(9 205)(10 206)(11 207)(12 208)(13 209)(14 210)(15 195)(16 196)(17 190)(18 191)(19 192)(20 193)(21 194)(22 212)(23 213)(24 214)(25 215)(26 216)(27 217)(28 211)(29 52)(30 53)(31 54)(32 55)(33 56)(34 50)(35 51)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 60)(44 61)(45 62)(46 63)(47 57)(48 58)(49 59)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(134 175)(135 169)(136 170)(137 171)(138 172)(139 173)(140 174)(141 163)(142 164)(143 165)(144 166)(145 167)(146 168)(147 162)(148 183)(149 184)(150 185)(151 186)(152 187)(153 188)(154 189)(155 176)(156 177)(157 178)(158 179)(159 180)(160 181)(161 182)(197 219)(198 220)(199 221)(200 222)(201 223)(202 224)(203 218)

G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,222)(9,223)(10,224)(11,218)(12,219)(13,220)(14,221)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,96)(86,97)(87,98)(88,92)(89,93)(90,94)(91,95)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,124)(114,125)(115,126)(116,120)(117,121)(118,122)(119,123)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,152)(142,153)(143,154)(144,148)(145,149)(146,150)(147,151)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,180)(170,181)(171,182)(172,176)(173,177)(174,178)(175,179)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,208)(198,209)(199,210)(200,204)(201,205)(202,206)(203,207), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,41)(30,42)(31,36)(32,37)(33,38)(34,39)(35,40)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102)(106,123)(107,124)(108,125)(109,126)(110,120)(111,121)(112,122)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,151)(135,152)(136,153)(137,154)(138,148)(139,149)(140,150)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,179)(163,180)(164,181)(165,182)(166,176)(167,177)(168,178)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,207)(191,208)(192,209)(193,210)(194,204)(195,205)(196,206)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,117)(16,118)(17,119)(18,113)(19,114)(20,115)(21,116)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,120)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,144)(44,145)(45,146)(46,147)(47,141)(48,142)(49,143)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (8,28)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,223)(16,224)(17,218)(18,219)(19,220)(20,221)(21,222)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(106,123)(107,124)(108,125)(109,126)(110,120)(111,121)(112,122)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,152)(142,153)(143,154)(144,148)(145,149)(146,150)(147,151)(190,207)(191,208)(192,209)(193,210)(194,204)(195,205)(196,206)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,78,34,102)(2,79,35,103)(3,80,29,104)(4,81,30,105)(5,82,31,99)(6,83,32,100)(7,84,33,101)(8,172,222,176)(9,173,223,177)(10,174,224,178)(11,175,218,179)(12,169,219,180)(13,170,220,181)(14,171,221,182)(15,184,22,167)(16,185,23,168)(17,186,24,162)(18,187,25,163)(19,188,26,164)(20,189,27,165)(21,183,28,166)(36,88,43,92)(37,89,44,93)(38,90,45,94)(39,91,46,95)(40,85,47,96)(41,86,48,97)(42,87,49,98)(50,130,74,106)(51,131,75,107)(52,132,76,108)(53,133,77,109)(54,127,71,110)(55,128,72,111)(56,129,73,112)(57,124,68,113)(58,125,69,114)(59,126,70,115)(60,120,64,116)(61,121,65,117)(62,122,66,118)(63,123,67,119)(134,203,158,207)(135,197,159,208)(136,198,160,209)(137,199,161,210)(138,200,155,204)(139,201,156,205)(140,202,157,206)(141,191,152,215)(142,192,153,216)(143,193,154,217)(144,194,148,211)(145,195,149,212)(146,196,150,213)(147,190,151,214), (1,74)(2,75)(3,76)(4,77)(5,71)(6,72)(7,73)(8,204)(9,205)(10,206)(11,207)(12,208)(13,209)(14,210)(15,195)(16,196)(17,190)(18,191)(19,192)(20,193)(21,194)(22,212)(23,213)(24,214)(25,215)(26,216)(27,217)(28,211)(29,52)(30,53)(31,54)(32,55)(33,56)(34,50)(35,51)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,60)(44,61)(45,62)(46,63)(47,57)(48,58)(49,59)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(134,175)(135,169)(136,170)(137,171)(138,172)(139,173)(140,174)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,162)(148,183)(149,184)(150,185)(151,186)(152,187)(153,188)(154,189)(155,176)(156,177)(157,178)(158,179)(159,180)(160,181)(161,182)(197,219)(198,220)(199,221)(200,222)(201,223)(202,224)(203,218)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,222)(9,223)(10,224)(11,218)(12,219)(13,220)(14,221)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,96)(86,97)(87,98)(88,92)(89,93)(90,94)(91,95)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,124)(114,125)(115,126)(116,120)(117,121)(118,122)(119,123)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,152)(142,153)(143,154)(144,148)(145,149)(146,150)(147,151)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,180)(170,181)(171,182)(172,176)(173,177)(174,178)(175,179)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,208)(198,209)(199,210)(200,204)(201,205)(202,206)(203,207), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,41)(30,42)(31,36)(32,37)(33,38)(34,39)(35,40)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102)(106,123)(107,124)(108,125)(109,126)(110,120)(111,121)(112,122)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,151)(135,152)(136,153)(137,154)(138,148)(139,149)(140,150)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,179)(163,180)(164,181)(165,182)(166,176)(167,177)(168,178)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,207)(191,208)(192,209)(193,210)(194,204)(195,205)(196,206)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,117)(16,118)(17,119)(18,113)(19,114)(20,115)(21,116)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,120)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,144)(44,145)(45,146)(46,147)(47,141)(48,142)(49,143)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (8,28)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,223)(16,224)(17,218)(18,219)(19,220)(20,221)(21,222)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(106,123)(107,124)(108,125)(109,126)(110,120)(111,121)(112,122)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,152)(142,153)(143,154)(144,148)(145,149)(146,150)(147,151)(190,207)(191,208)(192,209)(193,210)(194,204)(195,205)(196,206)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,78,34,102)(2,79,35,103)(3,80,29,104)(4,81,30,105)(5,82,31,99)(6,83,32,100)(7,84,33,101)(8,172,222,176)(9,173,223,177)(10,174,224,178)(11,175,218,179)(12,169,219,180)(13,170,220,181)(14,171,221,182)(15,184,22,167)(16,185,23,168)(17,186,24,162)(18,187,25,163)(19,188,26,164)(20,189,27,165)(21,183,28,166)(36,88,43,92)(37,89,44,93)(38,90,45,94)(39,91,46,95)(40,85,47,96)(41,86,48,97)(42,87,49,98)(50,130,74,106)(51,131,75,107)(52,132,76,108)(53,133,77,109)(54,127,71,110)(55,128,72,111)(56,129,73,112)(57,124,68,113)(58,125,69,114)(59,126,70,115)(60,120,64,116)(61,121,65,117)(62,122,66,118)(63,123,67,119)(134,203,158,207)(135,197,159,208)(136,198,160,209)(137,199,161,210)(138,200,155,204)(139,201,156,205)(140,202,157,206)(141,191,152,215)(142,192,153,216)(143,193,154,217)(144,194,148,211)(145,195,149,212)(146,196,150,213)(147,190,151,214), (1,74)(2,75)(3,76)(4,77)(5,71)(6,72)(7,73)(8,204)(9,205)(10,206)(11,207)(12,208)(13,209)(14,210)(15,195)(16,196)(17,190)(18,191)(19,192)(20,193)(21,194)(22,212)(23,213)(24,214)(25,215)(26,216)(27,217)(28,211)(29,52)(30,53)(31,54)(32,55)(33,56)(34,50)(35,51)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,60)(44,61)(45,62)(46,63)(47,57)(48,58)(49,59)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(134,175)(135,169)(136,170)(137,171)(138,172)(139,173)(140,174)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,162)(148,183)(149,184)(150,185)(151,186)(152,187)(153,188)(154,189)(155,176)(156,177)(157,178)(158,179)(159,180)(160,181)(161,182)(197,219)(198,220)(199,221)(200,222)(201,223)(202,224)(203,218) );

G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,222),(9,223),(10,224),(11,218),(12,219),(13,220),(14,221),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,68),(58,69),(59,70),(60,64),(61,65),(62,66),(63,67),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,96),(86,97),(87,98),(88,92),(89,93),(90,94),(91,95),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,124),(114,125),(115,126),(116,120),(117,121),(118,122),(119,123),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,152),(142,153),(143,154),(144,148),(145,149),(146,150),(147,151),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,180),(170,181),(171,182),(172,176),(173,177),(174,178),(175,179),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,208),(198,209),(199,210),(200,204),(201,205),(202,206),(203,207)], [(1,46),(2,47),(3,48),(4,49),(5,43),(6,44),(7,45),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,41),(30,42),(31,36),(32,37),(33,38),(34,39),(35,40),(50,67),(51,68),(52,69),(53,70),(54,64),(55,65),(56,66),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74),(78,95),(79,96),(80,97),(81,98),(82,92),(83,93),(84,94),(85,103),(86,104),(87,105),(88,99),(89,100),(90,101),(91,102),(106,123),(107,124),(108,125),(109,126),(110,120),(111,121),(112,122),(113,131),(114,132),(115,133),(116,127),(117,128),(118,129),(119,130),(134,151),(135,152),(136,153),(137,154),(138,148),(139,149),(140,150),(141,159),(142,160),(143,161),(144,155),(145,156),(146,157),(147,158),(162,179),(163,180),(164,181),(165,182),(166,176),(167,177),(168,178),(169,187),(170,188),(171,189),(172,183),(173,184),(174,185),(175,186),(190,207),(191,208),(192,209),(193,210),(194,204),(195,205),(196,206),(197,215),(198,216),(199,217),(200,211),(201,212),(202,213),(203,214)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,117),(16,118),(17,119),(18,113),(19,114),(20,115),(21,116),(22,121),(23,122),(24,123),(25,124),(26,125),(27,126),(28,120),(29,136),(30,137),(31,138),(32,139),(33,140),(34,134),(35,135),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,144),(44,145),(45,146),(46,147),(47,141),(48,142),(49,143),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(8,28),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,223),(16,224),(17,218),(18,219),(19,220),(20,221),(21,222),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,68),(58,69),(59,70),(60,64),(61,65),(62,66),(63,67),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105),(106,123),(107,124),(108,125),(109,126),(110,120),(111,121),(112,122),(113,131),(114,132),(115,133),(116,127),(117,128),(118,129),(119,130),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,152),(142,153),(143,154),(144,148),(145,149),(146,150),(147,151),(190,207),(191,208),(192,209),(193,210),(194,204),(195,205),(196,206),(197,215),(198,216),(199,217),(200,211),(201,212),(202,213),(203,214)], [(1,78,34,102),(2,79,35,103),(3,80,29,104),(4,81,30,105),(5,82,31,99),(6,83,32,100),(7,84,33,101),(8,172,222,176),(9,173,223,177),(10,174,224,178),(11,175,218,179),(12,169,219,180),(13,170,220,181),(14,171,221,182),(15,184,22,167),(16,185,23,168),(17,186,24,162),(18,187,25,163),(19,188,26,164),(20,189,27,165),(21,183,28,166),(36,88,43,92),(37,89,44,93),(38,90,45,94),(39,91,46,95),(40,85,47,96),(41,86,48,97),(42,87,49,98),(50,130,74,106),(51,131,75,107),(52,132,76,108),(53,133,77,109),(54,127,71,110),(55,128,72,111),(56,129,73,112),(57,124,68,113),(58,125,69,114),(59,126,70,115),(60,120,64,116),(61,121,65,117),(62,122,66,118),(63,123,67,119),(134,203,158,207),(135,197,159,208),(136,198,160,209),(137,199,161,210),(138,200,155,204),(139,201,156,205),(140,202,157,206),(141,191,152,215),(142,192,153,216),(143,193,154,217),(144,194,148,211),(145,195,149,212),(146,196,150,213),(147,190,151,214)], [(1,74),(2,75),(3,76),(4,77),(5,71),(6,72),(7,73),(8,204),(9,205),(10,206),(11,207),(12,208),(13,209),(14,210),(15,195),(16,196),(17,190),(18,191),(19,192),(20,193),(21,194),(22,212),(23,213),(24,214),(25,215),(26,216),(27,217),(28,211),(29,52),(30,53),(31,54),(32,55),(33,56),(34,50),(35,51),(36,64),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,60),(44,61),(45,62),(46,63),(47,57),(48,58),(49,59),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(85,113),(86,114),(87,115),(88,116),(89,117),(90,118),(91,119),(92,120),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,133),(134,175),(135,169),(136,170),(137,171),(138,172),(139,173),(140,174),(141,163),(142,164),(143,165),(144,166),(145,167),(146,168),(147,162),(148,183),(149,184),(150,185),(151,186),(152,187),(153,188),(154,189),(155,176),(156,177),(157,178),(158,179),(159,180),(160,181),(161,182),(197,219),(198,220),(199,221),(200,222),(201,223),(202,224),(203,218)]])

133 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4K 7A ··· 7F 14A ··· 14R 14S ··· 14AP 28A ··· 28BN order 1 2 2 2 2 2 2 2 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 4 4 4 4 ··· 4 1 ··· 1 1 ··· 1 4 ··· 4 4 ··· 4

133 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 2+ 1+4 2- 1+4 C7×2+ 1+4 C7×2- 1+4 kernel C7×C22.56C24 C7×C4⋊D4 C7×C22⋊Q8 C7×C22.D4 C7×C4.4D4 C7×C42.C2 C22.56C24 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C42.C2 C14 C14 C2 C2 # reps 1 4 4 4 2 1 6 24 24 24 12 6 2 1 12 6

Matrix representation of C7×C22.56C24 in GL8(𝔽29)

 24 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 23 21 0 0 0 0 0 0 8 6 0 0 0 0 0 0 0 0 6 8 0 0 0 0 0 0 21 23 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(29))| [24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[23,8,0,0,0,0,0,0,21,6,0,0,0,0,0,0,0,0,6,21,0,0,0,0,0,0,8,23,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C7×C22.56C24 in GAP, Magma, Sage, TeX

C_7\times C_2^2._{56}C_2^4
% in TeX

G:=Group("C7xC2^2.56C2^4");
// GroupNames label

G:=SmallGroup(448,1345);
// by ID

G=gap.SmallGroup(448,1345);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,3579,604,9635,1690]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=g^2=1,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
// generators/relations

׿
×
𝔽