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## G = C7×C22.53C24order 448 = 26·7

### Direct product of C7 and C22.53C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C22.53C24
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C22⋊C4 — C7×C4.4D4 — C7×C22.53C24
 Lower central C1 — C22 — C7×C22.53C24
 Upper central C1 — C2×C14 — C7×C22.53C24

Generators and relations for C7×C22.53C24
G = < a,b,c,d,e,f,g | a7=b2=c2=1, d2=f2=c, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, 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, C4×D4, C4×Q8, C22.D4, C4.4D4, C41D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22.53C24, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, D4×C28, Q8×C28, C7×C22.D4, C7×C4.4D4, C7×C41D4, C7×C22.53C24
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, 2+ 1+4, C22×C14, C22.53C24, C7×C4○D4, C23×C14, C14×C4○D4, C7×2+ 1+4, C7×C22.53C24

Smallest permutation representation of C7×C22.53C24
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)
(1 150 35 147)(2 151 29 141)(3 152 30 142)(4 153 31 143)(5 154 32 144)(6 148 33 145)(7 149 34 146)(8 130 18 112)(9 131 19 106)(10 132 20 107)(11 133 21 108)(12 127 15 109)(13 128 16 110)(14 129 17 111)(22 117 223 120)(23 118 224 121)(24 119 218 122)(25 113 219 123)(26 114 220 124)(27 115 221 125)(28 116 222 126)(36 138 44 156)(37 139 45 157)(38 140 46 158)(39 134 47 159)(40 135 48 160)(41 136 49 161)(42 137 43 155)(50 178 75 175)(51 179 76 169)(52 180 77 170)(53 181 71 171)(54 182 72 172)(55 176 73 173)(56 177 74 174)(57 188 67 163)(58 189 68 164)(59 183 69 165)(60 184 70 166)(61 185 64 167)(62 186 65 168)(63 187 66 162)(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)
(1 159 47 150)(2 160 48 151)(3 161 49 152)(4 155 43 153)(5 156 44 154)(6 157 45 148)(7 158 46 149)(8 118 224 112)(9 119 218 106)(10 113 219 107)(11 114 220 108)(12 115 221 109)(13 116 222 110)(14 117 223 111)(15 125 27 127)(16 126 28 128)(17 120 22 129)(18 121 23 130)(19 122 24 131)(20 123 25 132)(21 124 26 133)(29 135 40 141)(30 136 41 142)(31 137 42 143)(32 138 36 144)(33 139 37 145)(34 140 38 146)(35 134 39 147)(50 175 63 162)(51 169 57 163)(52 170 58 164)(53 171 59 165)(54 172 60 166)(55 173 61 167)(56 174 62 168)(64 185 73 176)(65 186 74 177)(66 187 75 178)(67 188 76 179)(68 189 77 180)(69 183 71 181)(70 184 72 182)(78 203 91 190)(79 197 85 191)(80 198 86 192)(81 199 87 193)(82 200 88 194)(83 201 89 195)(84 202 90 196)(92 213 101 204)(93 214 102 205)(94 215 103 206)(95 216 104 207)(96 217 105 208)(97 211 99 209)(98 212 100 210)
(1 91 35 94)(2 85 29 95)(3 86 30 96)(4 87 31 97)(5 88 32 98)(6 89 33 92)(7 90 34 93)(8 174 18 177)(9 175 19 178)(10 169 20 179)(11 170 21 180)(12 171 15 181)(13 172 16 182)(14 173 17 176)(22 185 223 167)(23 186 224 168)(24 187 218 162)(25 188 219 163)(26 189 220 164)(27 183 221 165)(28 184 222 166)(36 100 44 82)(37 101 45 83)(38 102 46 84)(39 103 47 78)(40 104 48 79)(41 105 49 80)(42 99 43 81)(50 131 75 106)(51 132 76 107)(52 133 77 108)(53 127 71 109)(54 128 72 110)(55 129 73 111)(56 130 74 112)(57 123 67 113)(58 124 68 114)(59 125 69 115)(60 126 70 116)(61 120 64 117)(62 121 65 118)(63 122 66 119)(134 215 159 190)(135 216 160 191)(136 217 161 192)(137 211 155 193)(138 212 156 194)(139 213 157 195)(140 214 158 196)(141 207 151 197)(142 208 152 198)(143 209 153 199)(144 210 154 200)(145 204 148 201)(146 205 149 202)(147 206 150 203)
(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), (1,150,35,147)(2,151,29,141)(3,152,30,142)(4,153,31,143)(5,154,32,144)(6,148,33,145)(7,149,34,146)(8,130,18,112)(9,131,19,106)(10,132,20,107)(11,133,21,108)(12,127,15,109)(13,128,16,110)(14,129,17,111)(22,117,223,120)(23,118,224,121)(24,119,218,122)(25,113,219,123)(26,114,220,124)(27,115,221,125)(28,116,222,126)(36,138,44,156)(37,139,45,157)(38,140,46,158)(39,134,47,159)(40,135,48,160)(41,136,49,161)(42,137,43,155)(50,178,75,175)(51,179,76,169)(52,180,77,170)(53,181,71,171)(54,182,72,172)(55,176,73,173)(56,177,74,174)(57,188,67,163)(58,189,68,164)(59,183,69,165)(60,184,70,166)(61,185,64,167)(62,186,65,168)(63,187,66,162)(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), (1,159,47,150)(2,160,48,151)(3,161,49,152)(4,155,43,153)(5,156,44,154)(6,157,45,148)(7,158,46,149)(8,118,224,112)(9,119,218,106)(10,113,219,107)(11,114,220,108)(12,115,221,109)(13,116,222,110)(14,117,223,111)(15,125,27,127)(16,126,28,128)(17,120,22,129)(18,121,23,130)(19,122,24,131)(20,123,25,132)(21,124,26,133)(29,135,40,141)(30,136,41,142)(31,137,42,143)(32,138,36,144)(33,139,37,145)(34,140,38,146)(35,134,39,147)(50,175,63,162)(51,169,57,163)(52,170,58,164)(53,171,59,165)(54,172,60,166)(55,173,61,167)(56,174,62,168)(64,185,73,176)(65,186,74,177)(66,187,75,178)(67,188,76,179)(68,189,77,180)(69,183,71,181)(70,184,72,182)(78,203,91,190)(79,197,85,191)(80,198,86,192)(81,199,87,193)(82,200,88,194)(83,201,89,195)(84,202,90,196)(92,213,101,204)(93,214,102,205)(94,215,103,206)(95,216,104,207)(96,217,105,208)(97,211,99,209)(98,212,100,210), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,174,18,177)(9,175,19,178)(10,169,20,179)(11,170,21,180)(12,171,15,181)(13,172,16,182)(14,173,17,176)(22,185,223,167)(23,186,224,168)(24,187,218,162)(25,188,219,163)(26,189,220,164)(27,183,221,165)(28,184,222,166)(36,100,44,82)(37,101,45,83)(38,102,46,84)(39,103,47,78)(40,104,48,79)(41,105,49,80)(42,99,43,81)(50,131,75,106)(51,132,76,107)(52,133,77,108)(53,127,71,109)(54,128,72,110)(55,129,73,111)(56,130,74,112)(57,123,67,113)(58,124,68,114)(59,125,69,115)(60,126,70,116)(61,120,64,117)(62,121,65,118)(63,122,66,119)(134,215,159,190)(135,216,160,191)(136,217,161,192)(137,211,155,193)(138,212,156,194)(139,213,157,195)(140,214,158,196)(141,207,151,197)(142,208,152,198)(143,209,153,199)(144,210,154,200)(145,204,148,201)(146,205,149,202)(147,206,150,203), (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), (1,150,35,147)(2,151,29,141)(3,152,30,142)(4,153,31,143)(5,154,32,144)(6,148,33,145)(7,149,34,146)(8,130,18,112)(9,131,19,106)(10,132,20,107)(11,133,21,108)(12,127,15,109)(13,128,16,110)(14,129,17,111)(22,117,223,120)(23,118,224,121)(24,119,218,122)(25,113,219,123)(26,114,220,124)(27,115,221,125)(28,116,222,126)(36,138,44,156)(37,139,45,157)(38,140,46,158)(39,134,47,159)(40,135,48,160)(41,136,49,161)(42,137,43,155)(50,178,75,175)(51,179,76,169)(52,180,77,170)(53,181,71,171)(54,182,72,172)(55,176,73,173)(56,177,74,174)(57,188,67,163)(58,189,68,164)(59,183,69,165)(60,184,70,166)(61,185,64,167)(62,186,65,168)(63,187,66,162)(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), (1,159,47,150)(2,160,48,151)(3,161,49,152)(4,155,43,153)(5,156,44,154)(6,157,45,148)(7,158,46,149)(8,118,224,112)(9,119,218,106)(10,113,219,107)(11,114,220,108)(12,115,221,109)(13,116,222,110)(14,117,223,111)(15,125,27,127)(16,126,28,128)(17,120,22,129)(18,121,23,130)(19,122,24,131)(20,123,25,132)(21,124,26,133)(29,135,40,141)(30,136,41,142)(31,137,42,143)(32,138,36,144)(33,139,37,145)(34,140,38,146)(35,134,39,147)(50,175,63,162)(51,169,57,163)(52,170,58,164)(53,171,59,165)(54,172,60,166)(55,173,61,167)(56,174,62,168)(64,185,73,176)(65,186,74,177)(66,187,75,178)(67,188,76,179)(68,189,77,180)(69,183,71,181)(70,184,72,182)(78,203,91,190)(79,197,85,191)(80,198,86,192)(81,199,87,193)(82,200,88,194)(83,201,89,195)(84,202,90,196)(92,213,101,204)(93,214,102,205)(94,215,103,206)(95,216,104,207)(96,217,105,208)(97,211,99,209)(98,212,100,210), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,174,18,177)(9,175,19,178)(10,169,20,179)(11,170,21,180)(12,171,15,181)(13,172,16,182)(14,173,17,176)(22,185,223,167)(23,186,224,168)(24,187,218,162)(25,188,219,163)(26,189,220,164)(27,183,221,165)(28,184,222,166)(36,100,44,82)(37,101,45,83)(38,102,46,84)(39,103,47,78)(40,104,48,79)(41,105,49,80)(42,99,43,81)(50,131,75,106)(51,132,76,107)(52,133,77,108)(53,127,71,109)(54,128,72,110)(55,129,73,111)(56,130,74,112)(57,123,67,113)(58,124,68,114)(59,125,69,115)(60,126,70,116)(61,120,64,117)(62,121,65,118)(63,122,66,119)(134,215,159,190)(135,216,160,191)(136,217,161,192)(137,211,155,193)(138,212,156,194)(139,213,157,195)(140,214,158,196)(141,207,151,197)(142,208,152,198)(143,209,153,199)(144,210,154,200)(145,204,148,201)(146,205,149,202)(147,206,150,203), (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)], [(1,150,35,147),(2,151,29,141),(3,152,30,142),(4,153,31,143),(5,154,32,144),(6,148,33,145),(7,149,34,146),(8,130,18,112),(9,131,19,106),(10,132,20,107),(11,133,21,108),(12,127,15,109),(13,128,16,110),(14,129,17,111),(22,117,223,120),(23,118,224,121),(24,119,218,122),(25,113,219,123),(26,114,220,124),(27,115,221,125),(28,116,222,126),(36,138,44,156),(37,139,45,157),(38,140,46,158),(39,134,47,159),(40,135,48,160),(41,136,49,161),(42,137,43,155),(50,178,75,175),(51,179,76,169),(52,180,77,170),(53,181,71,171),(54,182,72,172),(55,176,73,173),(56,177,74,174),(57,188,67,163),(58,189,68,164),(59,183,69,165),(60,184,70,166),(61,185,64,167),(62,186,65,168),(63,187,66,162),(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)], [(1,159,47,150),(2,160,48,151),(3,161,49,152),(4,155,43,153),(5,156,44,154),(6,157,45,148),(7,158,46,149),(8,118,224,112),(9,119,218,106),(10,113,219,107),(11,114,220,108),(12,115,221,109),(13,116,222,110),(14,117,223,111),(15,125,27,127),(16,126,28,128),(17,120,22,129),(18,121,23,130),(19,122,24,131),(20,123,25,132),(21,124,26,133),(29,135,40,141),(30,136,41,142),(31,137,42,143),(32,138,36,144),(33,139,37,145),(34,140,38,146),(35,134,39,147),(50,175,63,162),(51,169,57,163),(52,170,58,164),(53,171,59,165),(54,172,60,166),(55,173,61,167),(56,174,62,168),(64,185,73,176),(65,186,74,177),(66,187,75,178),(67,188,76,179),(68,189,77,180),(69,183,71,181),(70,184,72,182),(78,203,91,190),(79,197,85,191),(80,198,86,192),(81,199,87,193),(82,200,88,194),(83,201,89,195),(84,202,90,196),(92,213,101,204),(93,214,102,205),(94,215,103,206),(95,216,104,207),(96,217,105,208),(97,211,99,209),(98,212,100,210)], [(1,91,35,94),(2,85,29,95),(3,86,30,96),(4,87,31,97),(5,88,32,98),(6,89,33,92),(7,90,34,93),(8,174,18,177),(9,175,19,178),(10,169,20,179),(11,170,21,180),(12,171,15,181),(13,172,16,182),(14,173,17,176),(22,185,223,167),(23,186,224,168),(24,187,218,162),(25,188,219,163),(26,189,220,164),(27,183,221,165),(28,184,222,166),(36,100,44,82),(37,101,45,83),(38,102,46,84),(39,103,47,78),(40,104,48,79),(41,105,49,80),(42,99,43,81),(50,131,75,106),(51,132,76,107),(52,133,77,108),(53,127,71,109),(54,128,72,110),(55,129,73,111),(56,130,74,112),(57,123,67,113),(58,124,68,114),(59,125,69,115),(60,126,70,116),(61,120,64,117),(62,121,65,118),(63,122,66,119),(134,215,159,190),(135,216,160,191),(136,217,161,192),(137,211,155,193),(138,212,156,194),(139,213,157,195),(140,214,158,196),(141,207,151,197),(142,208,152,198),(143,209,153,199),(144,210,154,200),(145,204,148,201),(146,205,149,202),(147,206,150,203)], [(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×C22.53C24 D4×C28 Q8×C28 C7×C22.D4 C7×C4.4D4 C7×C4⋊1D4 C22.53C24 C4×D4 C4×Q8 C22.D4 C4.4D4 C4⋊1D4 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×C22.53C24 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 25 0 0 0 0 25
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 17 0 0 0 0 11 5 0 0 5 18
,
 17 0 0 0 0 17 0 0 0 0 18 7 0 0 24 11
,
 0 1 0 0 28 0 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 1 0 0 0 0 13 27 0 0 27 16
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,17,0,0,0,0,11,5,0,0,5,18],[17,0,0,0,0,17,0,0,0,0,18,24,0,0,7,11],[0,28,0,0,1,0,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,13,27,0,0,27,16] >;

C7×C22.53C24 in GAP, Magma, Sage, TeX

C_7\times C_2^2._{53}C_2^4
% in TeX

G:=Group("C7xC2^2.53C2^4");
// GroupNames label

G:=SmallGroup(448,1342);
// by ID

G=gap.SmallGroup(448,1342);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,2360,4790,1192,1690,416]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=1,d^2=f^2=c,e^2=c*b=b*c,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,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