Copied to
clipboard

## G = C2×D4⋊D15order 480 = 25·3·5

### Direct product of C2 and D4⋊D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×D4⋊D15
 Chief series C1 — C5 — C15 — C30 — C60 — D60 — C2×D60 — C2×D4⋊D15
 Lower central C15 — C30 — C60 — C2×D4⋊D15
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4⋊D15
G = < a,b,c,d,e | a2=b4=c2=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 980 in 152 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, D4 [×2], D4 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C15, C2×C8, D8 [×4], C2×D4, C2×D4, C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C3⋊C8 [×2], D12 [×3], C2×C12, C3×D4 [×2], C3×D4, C22×S3, C22×C6, D15 [×2], C30, C30 [×2], C30 [×2], C2×D8, C52C8 [×2], D20 [×3], C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×C10, C2×C3⋊C8, D4⋊S3 [×4], C2×D12, C6×D4, C60 [×2], D30 [×4], C2×C30, C2×C30 [×4], C2×C52C8, D4⋊D5 [×4], C2×D20, D4×C10, C2×D4⋊S3, C153C8 [×2], D60 [×2], D60, C2×C60, D4×C15 [×2], D4×C15, C22×D15, C22×C30, C2×D4⋊D5, C2×C153C8, D4⋊D15 [×4], C2×D60, D4×C30, C2×D4⋊D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C2×D8, C5⋊D4 [×2], C22×D5, D4⋊S3 [×2], C2×C3⋊D4, D30 [×3], D4⋊D5 [×2], C2×C5⋊D4, C2×D4⋊S3, C157D4 [×2], C22×D15, C2×D4⋊D5, D4⋊D15 [×2], C2×C157D4, C2×D4⋊D15

Smallest permutation representation of C2×D4⋊D15
On 240 points
Generators in S240
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 121)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 136)(26 137)(27 138)(28 139)(29 140)(30 141)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 164)(38 165)(39 151)(40 152)(41 153)(42 154)(43 155)(44 156)(45 157)(46 179)(47 180)(48 166)(49 167)(50 168)(51 169)(52 170)(53 171)(54 172)(55 173)(56 174)(57 175)(58 176)(59 177)(60 178)(61 191)(62 192)(63 193)(64 194)(65 195)(66 181)(67 182)(68 183)(69 184)(70 185)(71 186)(72 187)(73 188)(74 189)(75 190)(76 205)(77 206)(78 207)(79 208)(80 209)(81 210)(82 196)(83 197)(84 198)(85 199)(86 200)(87 201)(88 202)(89 203)(90 204)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 239)(107 240)(108 226)(109 227)(110 228)(111 229)(112 230)(113 231)(114 232)(115 233)(116 234)(117 235)(118 236)(119 237)(120 238)
(1 50 18 33)(2 51 19 34)(3 52 20 35)(4 53 21 36)(5 54 22 37)(6 55 23 38)(7 56 24 39)(8 57 25 40)(9 58 26 41)(10 59 27 42)(11 60 28 43)(12 46 29 44)(13 47 30 45)(14 48 16 31)(15 49 17 32)(61 98 86 111)(62 99 87 112)(63 100 88 113)(64 101 89 114)(65 102 90 115)(66 103 76 116)(67 104 77 117)(68 105 78 118)(69 91 79 119)(70 92 80 120)(71 93 81 106)(72 94 82 107)(73 95 83 108)(74 96 84 109)(75 97 85 110)(121 167 143 159)(122 168 144 160)(123 169 145 161)(124 170 146 162)(125 171 147 163)(126 172 148 164)(127 173 149 165)(128 174 150 151)(129 175 136 152)(130 176 137 153)(131 177 138 154)(132 178 139 155)(133 179 140 156)(134 180 141 157)(135 166 142 158)(181 223 205 234)(182 224 206 235)(183 225 207 236)(184 211 208 237)(185 212 209 238)(186 213 210 239)(187 214 196 240)(188 215 197 226)(189 216 198 227)(190 217 199 228)(191 218 200 229)(192 219 201 230)(193 220 202 231)(194 221 203 232)(195 222 204 233)
(1 228)(2 229)(3 230)(4 231)(5 232)(6 233)(7 234)(8 235)(9 236)(10 237)(11 238)(12 239)(13 240)(14 226)(15 227)(16 215)(17 216)(18 217)(19 218)(20 219)(21 220)(22 221)(23 222)(24 223)(25 224)(26 225)(27 211)(28 212)(29 213)(30 214)(31 188)(32 189)(33 190)(34 191)(35 192)(36 193)(37 194)(38 195)(39 181)(40 182)(41 183)(42 184)(43 185)(44 186)(45 187)(46 210)(47 196)(48 197)(49 198)(50 199)(51 200)(52 201)(53 202)(54 203)(55 204)(56 205)(57 206)(58 207)(59 208)(60 209)(61 161)(62 162)(63 163)(64 164)(65 165)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 174)(77 175)(78 176)(79 177)(80 178)(81 179)(82 180)(83 166)(84 167)(85 168)(86 169)(87 170)(88 171)(89 172)(90 173)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 136)(105 137)(106 133)(107 134)(108 135)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(118 130)(119 131)(120 132)
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)(226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 19)(17 18)(20 30)(21 29)(22 28)(23 27)(24 26)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(61 95)(62 94)(63 93)(64 92)(65 91)(66 105)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 118)(77 117)(78 116)(79 115)(80 114)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 120)(90 119)(121 122)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(151 176)(152 175)(153 174)(154 173)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 180)(163 179)(164 178)(165 177)(181 225)(182 224)(183 223)(184 222)(185 221)(186 220)(187 219)(188 218)(189 217)(190 216)(191 215)(192 214)(193 213)(194 212)(195 211)(196 230)(197 229)(198 228)(199 227)(200 226)(201 240)(202 239)(203 238)(204 237)(205 236)(206 235)(207 234)(208 233)(209 232)(210 231)

G:=sub<Sym(240)| (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,121)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,179)(47,180)(48,166)(49,167)(50,168)(51,169)(52,170)(53,171)(54,172)(55,173)(56,174)(57,175)(58,176)(59,177)(60,178)(61,191)(62,192)(63,193)(64,194)(65,195)(66,181)(67,182)(68,183)(69,184)(70,185)(71,186)(72,187)(73,188)(74,189)(75,190)(76,205)(77,206)(78,207)(79,208)(80,209)(81,210)(82,196)(83,197)(84,198)(85,199)(86,200)(87,201)(88,202)(89,203)(90,204)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,239)(107,240)(108,226)(109,227)(110,228)(111,229)(112,230)(113,231)(114,232)(115,233)(116,234)(117,235)(118,236)(119,237)(120,238), (1,50,18,33)(2,51,19,34)(3,52,20,35)(4,53,21,36)(5,54,22,37)(6,55,23,38)(7,56,24,39)(8,57,25,40)(9,58,26,41)(10,59,27,42)(11,60,28,43)(12,46,29,44)(13,47,30,45)(14,48,16,31)(15,49,17,32)(61,98,86,111)(62,99,87,112)(63,100,88,113)(64,101,89,114)(65,102,90,115)(66,103,76,116)(67,104,77,117)(68,105,78,118)(69,91,79,119)(70,92,80,120)(71,93,81,106)(72,94,82,107)(73,95,83,108)(74,96,84,109)(75,97,85,110)(121,167,143,159)(122,168,144,160)(123,169,145,161)(124,170,146,162)(125,171,147,163)(126,172,148,164)(127,173,149,165)(128,174,150,151)(129,175,136,152)(130,176,137,153)(131,177,138,154)(132,178,139,155)(133,179,140,156)(134,180,141,157)(135,166,142,158)(181,223,205,234)(182,224,206,235)(183,225,207,236)(184,211,208,237)(185,212,209,238)(186,213,210,239)(187,214,196,240)(188,215,197,226)(189,216,198,227)(190,217,199,228)(191,218,200,229)(192,219,201,230)(193,220,202,231)(194,221,203,232)(195,222,204,233), (1,228)(2,229)(3,230)(4,231)(5,232)(6,233)(7,234)(8,235)(9,236)(10,237)(11,238)(12,239)(13,240)(14,226)(15,227)(16,215)(17,216)(18,217)(19,218)(20,219)(21,220)(22,221)(23,222)(24,223)(25,224)(26,225)(27,211)(28,212)(29,213)(30,214)(31,188)(32,189)(33,190)(34,191)(35,192)(36,193)(37,194)(38,195)(39,181)(40,182)(41,183)(42,184)(43,185)(44,186)(45,187)(46,210)(47,196)(48,197)(49,198)(50,199)(51,200)(52,201)(53,202)(54,203)(55,204)(56,205)(57,206)(58,207)(59,208)(60,209)(61,161)(62,162)(63,163)(64,164)(65,165)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,174)(77,175)(78,176)(79,177)(80,178)(81,179)(82,180)(83,166)(84,167)(85,168)(86,169)(87,170)(88,171)(89,172)(90,173)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,136)(105,137)(106,133)(107,134)(108,135)(109,121)(110,122)(111,123)(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,19)(17,18)(20,30)(21,29)(22,28)(23,27)(24,26)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(61,95)(62,94)(63,93)(64,92)(65,91)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,120)(90,119)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(151,176)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,180)(163,179)(164,178)(165,177)(181,225)(182,224)(183,223)(184,222)(185,221)(186,220)(187,219)(188,218)(189,217)(190,216)(191,215)(192,214)(193,213)(194,212)(195,211)(196,230)(197,229)(198,228)(199,227)(200,226)(201,240)(202,239)(203,238)(204,237)(205,236)(206,235)(207,234)(208,233)(209,232)(210,231)>;

G:=Group( (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,121)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,179)(47,180)(48,166)(49,167)(50,168)(51,169)(52,170)(53,171)(54,172)(55,173)(56,174)(57,175)(58,176)(59,177)(60,178)(61,191)(62,192)(63,193)(64,194)(65,195)(66,181)(67,182)(68,183)(69,184)(70,185)(71,186)(72,187)(73,188)(74,189)(75,190)(76,205)(77,206)(78,207)(79,208)(80,209)(81,210)(82,196)(83,197)(84,198)(85,199)(86,200)(87,201)(88,202)(89,203)(90,204)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,239)(107,240)(108,226)(109,227)(110,228)(111,229)(112,230)(113,231)(114,232)(115,233)(116,234)(117,235)(118,236)(119,237)(120,238), (1,50,18,33)(2,51,19,34)(3,52,20,35)(4,53,21,36)(5,54,22,37)(6,55,23,38)(7,56,24,39)(8,57,25,40)(9,58,26,41)(10,59,27,42)(11,60,28,43)(12,46,29,44)(13,47,30,45)(14,48,16,31)(15,49,17,32)(61,98,86,111)(62,99,87,112)(63,100,88,113)(64,101,89,114)(65,102,90,115)(66,103,76,116)(67,104,77,117)(68,105,78,118)(69,91,79,119)(70,92,80,120)(71,93,81,106)(72,94,82,107)(73,95,83,108)(74,96,84,109)(75,97,85,110)(121,167,143,159)(122,168,144,160)(123,169,145,161)(124,170,146,162)(125,171,147,163)(126,172,148,164)(127,173,149,165)(128,174,150,151)(129,175,136,152)(130,176,137,153)(131,177,138,154)(132,178,139,155)(133,179,140,156)(134,180,141,157)(135,166,142,158)(181,223,205,234)(182,224,206,235)(183,225,207,236)(184,211,208,237)(185,212,209,238)(186,213,210,239)(187,214,196,240)(188,215,197,226)(189,216,198,227)(190,217,199,228)(191,218,200,229)(192,219,201,230)(193,220,202,231)(194,221,203,232)(195,222,204,233), (1,228)(2,229)(3,230)(4,231)(5,232)(6,233)(7,234)(8,235)(9,236)(10,237)(11,238)(12,239)(13,240)(14,226)(15,227)(16,215)(17,216)(18,217)(19,218)(20,219)(21,220)(22,221)(23,222)(24,223)(25,224)(26,225)(27,211)(28,212)(29,213)(30,214)(31,188)(32,189)(33,190)(34,191)(35,192)(36,193)(37,194)(38,195)(39,181)(40,182)(41,183)(42,184)(43,185)(44,186)(45,187)(46,210)(47,196)(48,197)(49,198)(50,199)(51,200)(52,201)(53,202)(54,203)(55,204)(56,205)(57,206)(58,207)(59,208)(60,209)(61,161)(62,162)(63,163)(64,164)(65,165)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,174)(77,175)(78,176)(79,177)(80,178)(81,179)(82,180)(83,166)(84,167)(85,168)(86,169)(87,170)(88,171)(89,172)(90,173)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,136)(105,137)(106,133)(107,134)(108,135)(109,121)(110,122)(111,123)(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,19)(17,18)(20,30)(21,29)(22,28)(23,27)(24,26)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(61,95)(62,94)(63,93)(64,92)(65,91)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,118)(77,117)(78,116)(79,115)(80,114)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,120)(90,119)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(151,176)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,180)(163,179)(164,178)(165,177)(181,225)(182,224)(183,223)(184,222)(185,221)(186,220)(187,219)(188,218)(189,217)(190,216)(191,215)(192,214)(193,213)(194,212)(195,211)(196,230)(197,229)(198,228)(199,227)(200,226)(201,240)(202,239)(203,238)(204,237)(205,236)(206,235)(207,234)(208,233)(209,232)(210,231) );

G=PermutationGroup([(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,131),(11,132),(12,133),(13,134),(14,135),(15,121),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,136),(26,137),(27,138),(28,139),(29,140),(30,141),(31,158),(32,159),(33,160),(34,161),(35,162),(36,163),(37,164),(38,165),(39,151),(40,152),(41,153),(42,154),(43,155),(44,156),(45,157),(46,179),(47,180),(48,166),(49,167),(50,168),(51,169),(52,170),(53,171),(54,172),(55,173),(56,174),(57,175),(58,176),(59,177),(60,178),(61,191),(62,192),(63,193),(64,194),(65,195),(66,181),(67,182),(68,183),(69,184),(70,185),(71,186),(72,187),(73,188),(74,189),(75,190),(76,205),(77,206),(78,207),(79,208),(80,209),(81,210),(82,196),(83,197),(84,198),(85,199),(86,200),(87,201),(88,202),(89,203),(90,204),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,225),(106,239),(107,240),(108,226),(109,227),(110,228),(111,229),(112,230),(113,231),(114,232),(115,233),(116,234),(117,235),(118,236),(119,237),(120,238)], [(1,50,18,33),(2,51,19,34),(3,52,20,35),(4,53,21,36),(5,54,22,37),(6,55,23,38),(7,56,24,39),(8,57,25,40),(9,58,26,41),(10,59,27,42),(11,60,28,43),(12,46,29,44),(13,47,30,45),(14,48,16,31),(15,49,17,32),(61,98,86,111),(62,99,87,112),(63,100,88,113),(64,101,89,114),(65,102,90,115),(66,103,76,116),(67,104,77,117),(68,105,78,118),(69,91,79,119),(70,92,80,120),(71,93,81,106),(72,94,82,107),(73,95,83,108),(74,96,84,109),(75,97,85,110),(121,167,143,159),(122,168,144,160),(123,169,145,161),(124,170,146,162),(125,171,147,163),(126,172,148,164),(127,173,149,165),(128,174,150,151),(129,175,136,152),(130,176,137,153),(131,177,138,154),(132,178,139,155),(133,179,140,156),(134,180,141,157),(135,166,142,158),(181,223,205,234),(182,224,206,235),(183,225,207,236),(184,211,208,237),(185,212,209,238),(186,213,210,239),(187,214,196,240),(188,215,197,226),(189,216,198,227),(190,217,199,228),(191,218,200,229),(192,219,201,230),(193,220,202,231),(194,221,203,232),(195,222,204,233)], [(1,228),(2,229),(3,230),(4,231),(5,232),(6,233),(7,234),(8,235),(9,236),(10,237),(11,238),(12,239),(13,240),(14,226),(15,227),(16,215),(17,216),(18,217),(19,218),(20,219),(21,220),(22,221),(23,222),(24,223),(25,224),(26,225),(27,211),(28,212),(29,213),(30,214),(31,188),(32,189),(33,190),(34,191),(35,192),(36,193),(37,194),(38,195),(39,181),(40,182),(41,183),(42,184),(43,185),(44,186),(45,187),(46,210),(47,196),(48,197),(49,198),(50,199),(51,200),(52,201),(53,202),(54,203),(55,204),(56,205),(57,206),(58,207),(59,208),(60,209),(61,161),(62,162),(63,163),(64,164),(65,165),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,174),(77,175),(78,176),(79,177),(80,178),(81,179),(82,180),(83,166),(84,167),(85,168),(86,169),(87,170),(88,171),(89,172),(90,173),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143),(97,144),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,136),(105,137),(106,133),(107,134),(108,135),(109,121),(110,122),(111,123),(112,124),(113,125),(114,126),(115,127),(116,128),(117,129),(118,130),(119,131),(120,132)], [(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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225),(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,19),(17,18),(20,30),(21,29),(22,28),(23,27),(24,26),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,54),(44,53),(45,52),(61,95),(62,94),(63,93),(64,92),(65,91),(66,105),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(73,98),(74,97),(75,96),(76,118),(77,117),(78,116),(79,115),(80,114),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,120),(90,119),(121,122),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(137,150),(138,149),(139,148),(140,147),(141,146),(142,145),(143,144),(151,176),(152,175),(153,174),(154,173),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,180),(163,179),(164,178),(165,177),(181,225),(182,224),(183,223),(184,222),(185,221),(186,220),(187,219),(188,218),(189,217),(190,216),(191,215),(192,214),(193,213),(194,212),(195,211),(196,230),(197,229),(198,228),(199,227),(200,226),(201,240),(202,239),(203,238),(204,237),(205,236),(206,235),(207,234),(208,233),(209,232),(210,231)])

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 4 60 60 2 2 2 2 2 2 2 2 4 4 4 4 30 30 30 30 2 ··· 2 4 ··· 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D8 D10 D10 C3⋊D4 C3⋊D4 D15 C5⋊D4 C5⋊D4 D30 D30 C15⋊7D4 C15⋊7D4 D4⋊S3 D4⋊D5 D4⋊D15 kernel C2×D4⋊D15 C2×C15⋊3C8 D4⋊D15 C2×D60 D4×C30 D4×C10 C60 C2×C30 C6×D4 C2×C20 C5×D4 C30 C2×C12 C3×D4 C20 C2×C10 C2×D4 C12 C2×C6 C2×C4 D4 C4 C22 C10 C6 C2 # reps 1 1 4 1 1 1 1 1 2 1 2 4 2 4 2 2 4 4 4 4 8 8 8 2 4 8

Matrix representation of C2×D4⋊D15 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240
,
 1 0 0 0 0 1 0 0 0 0 240 192 0 0 123 1
,
 240 0 0 0 0 240 0 0 0 0 22 57 0 0 93 219
,
 110 173 0 0 215 84 0 0 0 0 1 0 0 0 0 1
,
 80 64 0 0 77 161 0 0 0 0 1 0 0 0 118 240
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,240,123,0,0,192,1],[240,0,0,0,0,240,0,0,0,0,22,93,0,0,57,219],[110,215,0,0,173,84,0,0,0,0,1,0,0,0,0,1],[80,77,0,0,64,161,0,0,0,0,1,118,0,0,0,240] >;

C2×D4⋊D15 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_{15}
% in TeX

G:=Group("C2xD4:D15");
// GroupNames label

G:=SmallGroup(480,896);
// by ID

G=gap.SmallGroup(480,896);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,675,185,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^15=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽