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## G = C7×C22.49C24order 448 = 26·7

### Direct product of C7 and C22.49C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C22.49C24
 Chief series C1 — C2 — C22 — C2×C14 — C22×C14 — C7×C22⋊C4 — C7×C4.4D4 — C7×C22.49C24
 Lower central C1 — C22 — C7×C22.49C24
 Upper central C1 — C2×C14 — C7×C22.49C24

Generators and relations for C7×C22.49C24
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=c, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 362 in 236 conjugacy classes, 150 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22.49C24, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C42⋊C2, D4×C28, C7×C4⋊D4, C7×C4.4D4, C7×C4⋊Q8, C7×C22.49C24
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, 2+ 1+4, C22×C14, C22.49C24, C7×C4○D4, C23×C14, C14×C4○D4, C7×2+ 1+4, C7×C22.49C24

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

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

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

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

175 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4Q 7A ··· 7F 14A ··· 14R 14S ··· 14AP 28A ··· 28BT 28BU ··· 28CX order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 4 4 2 ··· 2 4 ··· 4 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

175 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C4○D4 C7×C4○D4 2+ 1+4 C7×2+ 1+4 kernel C7×C22.49C24 C7×C42⋊C2 D4×C28 C7×C4⋊D4 C7×C4.4D4 C7×C4⋊Q8 C22.49C24 C42⋊C2 C4×D4 C4⋊D4 C4.4D4 C4⋊Q8 C28 C4 C14 C2 # reps 1 4 2 4 4 1 6 24 12 24 24 6 8 48 1 6

Matrix representation of C7×C22.49C24 in GL4(𝔽29) generated by

 23 0 0 0 0 23 0 0 0 0 1 0 0 0 0 1
,
 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 0 17 0 0 12 0 0 0 0 0 0 12 0 0 17 0
,
 1 0 0 0 0 28 0 0 0 0 17 0 0 0 0 17
,
 12 0 0 0 0 12 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 28 0 0 0 0 0 28 0 0 0 0 28
G:=sub<GL(4,GF(29))| [23,0,0,0,0,23,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[0,12,0,0,17,0,0,0,0,0,0,17,0,0,12,0],[1,0,0,0,0,28,0,0,0,0,17,0,0,0,0,17],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0],[0,28,0,0,1,0,0,0,0,0,28,0,0,0,0,28] >;

C7×C22.49C24 in GAP, Magma, Sage, TeX

C_7\times C_2^2._{49}C_2^4
% in TeX

G:=Group("C7xC2^2.49C2^4");
// GroupNames label

G:=SmallGroup(448,1338);
// by ID

G=gap.SmallGroup(448,1338);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1576,4790,604,1690,416]);
// Polycyclic

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

׿
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𝔽