direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊3D15, Q8⋊6D30, D60⋊26C22, C60.86C23, C30.62C24, D30.27C23, Dic15.46C23, (C6×Q8)⋊6D5, (C5×Q8)⋊22D6, (C2×Q8)⋊8D15, (Q8×C30)⋊6C2, (C2×D60)⋊15C2, (C3×Q8)⋊19D10, (Q8×C10)⋊10S3, (C2×C4).62D30, C30⋊17(C4○D4), C6⋊3(Q8⋊2D5), (C2×C20).172D6, C6.62(C23×D5), C10⋊3(Q8⋊3S3), (C2×C12).170D10, (C4×D15)⋊18C22, C10.62(S3×C23), (C2×C60).88C22, (Q8×C15)⋊21C22, C2.10(C23×D15), C4.23(C22×D15), C20.136(C22×S3), (C2×C30).323C23, C12.134(C22×D5), C22.32(C22×D15), (C22×D15).92C22, (C2×Dic15).241C22, (C2×C4×D15)⋊5C2, C15⋊26(C2×C4○D4), C5⋊4(C2×Q8⋊3S3), C3⋊4(C2×Q8⋊2D5), (C2×C6).319(C22×D5), (C2×C10).319(C22×S3), SmallGroup(480,1173)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1876 in 328 conjugacy classes, 127 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×6], C4 [×2], C22, C22 [×12], C5, S3 [×6], C6, C6 [×2], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10, C10 [×2], Dic3 [×2], C12 [×6], D6 [×12], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×6], D10 [×12], C2×C10, C4×S3 [×12], D12 [×12], C2×Dic3, C2×C12 [×3], C3×Q8 [×4], C22×S3 [×3], D15 [×6], C30, C30 [×2], C2×C4○D4, C4×D5 [×12], D20 [×12], C2×Dic5, C2×C20 [×3], C5×Q8 [×4], C22×D5 [×3], S3×C2×C4 [×3], C2×D12 [×3], Q8⋊3S3 [×8], C6×Q8, Dic15 [×2], C60 [×6], D30 [×6], D30 [×6], C2×C30, C2×C4×D5 [×3], C2×D20 [×3], Q8⋊2D5 [×8], Q8×C10, C2×Q8⋊3S3, C4×D15 [×12], D60 [×12], C2×Dic15, C2×C60 [×3], Q8×C15 [×4], C22×D15 [×3], C2×Q8⋊2D5, C2×C4×D15 [×3], C2×D60 [×3], Q8⋊3D15 [×8], Q8×C30, C2×Q8⋊3D15
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], D15, C2×C4○D4, C22×D5 [×7], Q8⋊3S3 [×2], S3×C23, D30 [×7], Q8⋊2D5 [×2], C23×D5, C2×Q8⋊3S3, C22×D15 [×7], C2×Q8⋊2D5, Q8⋊3D15 [×2], C23×D15, C2×Q8⋊3D15
Generators and relations
G = < a,b,c,d,e | a2=b4=d15=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
►On 240 points
Generators in S240
(1 136)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 166)(42 167)(43 168)(44 169)(45 170)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 161)(53 162)(54 163)(55 164)(56 165)(57 151)(58 152)(59 153)(60 154)(61 209)(62 210)(63 196)(64 197)(65 198)(66 199)(67 200)(68 201)(69 202)(70 203)(71 204)(72 205)(73 206)(74 207)(75 208)(76 194)(77 195)(78 181)(79 182)(80 183)(81 184)(82 185)(83 186)(84 187)(85 188)(86 189)(87 190)(88 191)(89 192)(90 193)(91 226)(92 227)(93 228)(94 229)(95 230)(96 231)(97 232)(98 233)(99 234)(100 235)(101 236)(102 237)(103 238)(104 239)(105 240)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)(113 218)(114 219)(115 220)(116 221)(117 222)(118 223)(119 224)(120 225)
(1 81 18 74)(2 82 19 75)(3 83 20 61)(4 84 21 62)(5 85 22 63)(6 86 23 64)(7 87 24 65)(8 88 25 66)(9 89 26 67)(10 90 27 68)(11 76 28 69)(12 77 29 70)(13 78 30 71)(14 79 16 72)(15 80 17 73)(31 118 51 91)(32 119 52 92)(33 120 53 93)(34 106 54 94)(35 107 55 95)(36 108 56 96)(37 109 57 97)(38 110 58 98)(39 111 59 99)(40 112 60 100)(41 113 46 101)(42 114 47 102)(43 115 48 103)(44 116 49 104)(45 117 50 105)(121 199 143 191)(122 200 144 192)(123 201 145 193)(124 202 146 194)(125 203 147 195)(126 204 148 181)(127 205 149 182)(128 206 150 183)(129 207 136 184)(130 208 137 185)(131 209 138 186)(132 210 139 187)(133 196 140 188)(134 197 141 189)(135 198 142 190)(151 232 177 214)(152 233 178 215)(153 234 179 216)(154 235 180 217)(155 236 166 218)(156 237 167 219)(157 238 168 220)(158 239 169 221)(159 240 170 222)(160 226 171 223)(161 227 172 224)(162 228 173 225)(163 229 174 211)(164 230 175 212)(165 231 176 213)
(1 236 18 218)(2 237 19 219)(3 238 20 220)(4 239 21 221)(5 240 22 222)(6 226 23 223)(7 227 24 224)(8 228 25 225)(9 229 26 211)(10 230 27 212)(11 231 28 213)(12 232 29 214)(13 233 30 215)(14 234 16 216)(15 235 17 217)(31 189 51 197)(32 190 52 198)(33 191 53 199)(34 192 54 200)(35 193 55 201)(36 194 56 202)(37 195 57 203)(38 181 58 204)(39 182 59 205)(40 183 60 206)(41 184 46 207)(42 185 47 208)(43 186 48 209)(44 187 49 210)(45 188 50 196)(61 168 83 157)(62 169 84 158)(63 170 85 159)(64 171 86 160)(65 172 87 161)(66 173 88 162)(67 174 89 163)(68 175 90 164)(69 176 76 165)(70 177 77 151)(71 178 78 152)(72 179 79 153)(73 180 80 154)(74 166 81 155)(75 167 82 156)(91 134 118 141)(92 135 119 142)(93 121 120 143)(94 122 106 144)(95 123 107 145)(96 124 108 146)(97 125 109 147)(98 126 110 148)(99 127 111 149)(100 128 112 150)(101 129 113 136)(102 130 114 137)(103 131 115 138)(104 132 116 139)(105 133 117 140)
(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)
(1 183)(2 182)(3 181)(4 195)(5 194)(6 193)(7 192)(8 191)(9 190)(10 189)(11 188)(12 187)(13 186)(14 185)(15 184)(16 208)(17 207)(18 206)(19 205)(20 204)(21 203)(22 202)(23 201)(24 200)(25 199)(26 198)(27 197)(28 196)(29 210)(30 209)(31 212)(32 211)(33 225)(34 224)(35 223)(36 222)(37 221)(38 220)(39 219)(40 218)(41 217)(42 216)(43 215)(44 214)(45 213)(46 235)(47 234)(48 233)(49 232)(50 231)(51 230)(52 229)(53 228)(54 227)(55 226)(56 240)(57 239)(58 238)(59 237)(60 236)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 135)(68 134)(69 133)(70 132)(71 131)(72 130)(73 129)(74 128)(75 127)(76 140)(77 139)(78 138)(79 137)(80 136)(81 150)(82 149)(83 148)(84 147)(85 146)(86 145)(87 144)(88 143)(89 142)(90 141)(91 164)(92 163)(93 162)(94 161)(95 160)(96 159)(97 158)(98 157)(99 156)(100 155)(101 154)(102 153)(103 152)(104 151)(105 165)(106 172)(107 171)(108 170)(109 169)(110 168)(111 167)(112 166)(113 180)(114 179)(115 178)(116 177)(117 176)(118 175)(119 174)(120 173)
G:=sub<Sym(240)| (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,166)(42,167)(43,168)(44,169)(45,170)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,165)(57,151)(58,152)(59,153)(60,154)(61,209)(62,210)(63,196)(64,197)(65,198)(66,199)(67,200)(68,201)(69,202)(70,203)(71,204)(72,205)(73,206)(74,207)(75,208)(76,194)(77,195)(78,181)(79,182)(80,183)(81,184)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(89,192)(90,193)(91,226)(92,227)(93,228)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225), (1,81,18,74)(2,82,19,75)(3,83,20,61)(4,84,21,62)(5,85,22,63)(6,86,23,64)(7,87,24,65)(8,88,25,66)(9,89,26,67)(10,90,27,68)(11,76,28,69)(12,77,29,70)(13,78,30,71)(14,79,16,72)(15,80,17,73)(31,118,51,91)(32,119,52,92)(33,120,53,93)(34,106,54,94)(35,107,55,95)(36,108,56,96)(37,109,57,97)(38,110,58,98)(39,111,59,99)(40,112,60,100)(41,113,46,101)(42,114,47,102)(43,115,48,103)(44,116,49,104)(45,117,50,105)(121,199,143,191)(122,200,144,192)(123,201,145,193)(124,202,146,194)(125,203,147,195)(126,204,148,181)(127,205,149,182)(128,206,150,183)(129,207,136,184)(130,208,137,185)(131,209,138,186)(132,210,139,187)(133,196,140,188)(134,197,141,189)(135,198,142,190)(151,232,177,214)(152,233,178,215)(153,234,179,216)(154,235,180,217)(155,236,166,218)(156,237,167,219)(157,238,168,220)(158,239,169,221)(159,240,170,222)(160,226,171,223)(161,227,172,224)(162,228,173,225)(163,229,174,211)(164,230,175,212)(165,231,176,213), (1,236,18,218)(2,237,19,219)(3,238,20,220)(4,239,21,221)(5,240,22,222)(6,226,23,223)(7,227,24,224)(8,228,25,225)(9,229,26,211)(10,230,27,212)(11,231,28,213)(12,232,29,214)(13,233,30,215)(14,234,16,216)(15,235,17,217)(31,189,51,197)(32,190,52,198)(33,191,53,199)(34,192,54,200)(35,193,55,201)(36,194,56,202)(37,195,57,203)(38,181,58,204)(39,182,59,205)(40,183,60,206)(41,184,46,207)(42,185,47,208)(43,186,48,209)(44,187,49,210)(45,188,50,196)(61,168,83,157)(62,169,84,158)(63,170,85,159)(64,171,86,160)(65,172,87,161)(66,173,88,162)(67,174,89,163)(68,175,90,164)(69,176,76,165)(70,177,77,151)(71,178,78,152)(72,179,79,153)(73,180,80,154)(74,166,81,155)(75,167,82,156)(91,134,118,141)(92,135,119,142)(93,121,120,143)(94,122,106,144)(95,123,107,145)(96,124,108,146)(97,125,109,147)(98,126,110,148)(99,127,111,149)(100,128,112,150)(101,129,113,136)(102,130,114,137)(103,131,115,138)(104,132,116,139)(105,133,117,140), (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), (1,183)(2,182)(3,181)(4,195)(5,194)(6,193)(7,192)(8,191)(9,190)(10,189)(11,188)(12,187)(13,186)(14,185)(15,184)(16,208)(17,207)(18,206)(19,205)(20,204)(21,203)(22,202)(23,201)(24,200)(25,199)(26,198)(27,197)(28,196)(29,210)(30,209)(31,212)(32,211)(33,225)(34,224)(35,223)(36,222)(37,221)(38,220)(39,219)(40,218)(41,217)(42,216)(43,215)(44,214)(45,213)(46,235)(47,234)(48,233)(49,232)(50,231)(51,230)(52,229)(53,228)(54,227)(55,226)(56,240)(57,239)(58,238)(59,237)(60,236)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,135)(68,134)(69,133)(70,132)(71,131)(72,130)(73,129)(74,128)(75,127)(76,140)(77,139)(78,138)(79,137)(80,136)(81,150)(82,149)(83,148)(84,147)(85,146)(86,145)(87,144)(88,143)(89,142)(90,141)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,165)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)(113,180)(114,179)(115,178)(116,177)(117,176)(118,175)(119,174)(120,173)>;
G:=Group( (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,166)(42,167)(43,168)(44,169)(45,170)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,165)(57,151)(58,152)(59,153)(60,154)(61,209)(62,210)(63,196)(64,197)(65,198)(66,199)(67,200)(68,201)(69,202)(70,203)(71,204)(72,205)(73,206)(74,207)(75,208)(76,194)(77,195)(78,181)(79,182)(80,183)(81,184)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(89,192)(90,193)(91,226)(92,227)(93,228)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225), (1,81,18,74)(2,82,19,75)(3,83,20,61)(4,84,21,62)(5,85,22,63)(6,86,23,64)(7,87,24,65)(8,88,25,66)(9,89,26,67)(10,90,27,68)(11,76,28,69)(12,77,29,70)(13,78,30,71)(14,79,16,72)(15,80,17,73)(31,118,51,91)(32,119,52,92)(33,120,53,93)(34,106,54,94)(35,107,55,95)(36,108,56,96)(37,109,57,97)(38,110,58,98)(39,111,59,99)(40,112,60,100)(41,113,46,101)(42,114,47,102)(43,115,48,103)(44,116,49,104)(45,117,50,105)(121,199,143,191)(122,200,144,192)(123,201,145,193)(124,202,146,194)(125,203,147,195)(126,204,148,181)(127,205,149,182)(128,206,150,183)(129,207,136,184)(130,208,137,185)(131,209,138,186)(132,210,139,187)(133,196,140,188)(134,197,141,189)(135,198,142,190)(151,232,177,214)(152,233,178,215)(153,234,179,216)(154,235,180,217)(155,236,166,218)(156,237,167,219)(157,238,168,220)(158,239,169,221)(159,240,170,222)(160,226,171,223)(161,227,172,224)(162,228,173,225)(163,229,174,211)(164,230,175,212)(165,231,176,213), (1,236,18,218)(2,237,19,219)(3,238,20,220)(4,239,21,221)(5,240,22,222)(6,226,23,223)(7,227,24,224)(8,228,25,225)(9,229,26,211)(10,230,27,212)(11,231,28,213)(12,232,29,214)(13,233,30,215)(14,234,16,216)(15,235,17,217)(31,189,51,197)(32,190,52,198)(33,191,53,199)(34,192,54,200)(35,193,55,201)(36,194,56,202)(37,195,57,203)(38,181,58,204)(39,182,59,205)(40,183,60,206)(41,184,46,207)(42,185,47,208)(43,186,48,209)(44,187,49,210)(45,188,50,196)(61,168,83,157)(62,169,84,158)(63,170,85,159)(64,171,86,160)(65,172,87,161)(66,173,88,162)(67,174,89,163)(68,175,90,164)(69,176,76,165)(70,177,77,151)(71,178,78,152)(72,179,79,153)(73,180,80,154)(74,166,81,155)(75,167,82,156)(91,134,118,141)(92,135,119,142)(93,121,120,143)(94,122,106,144)(95,123,107,145)(96,124,108,146)(97,125,109,147)(98,126,110,148)(99,127,111,149)(100,128,112,150)(101,129,113,136)(102,130,114,137)(103,131,115,138)(104,132,116,139)(105,133,117,140), (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), (1,183)(2,182)(3,181)(4,195)(5,194)(6,193)(7,192)(8,191)(9,190)(10,189)(11,188)(12,187)(13,186)(14,185)(15,184)(16,208)(17,207)(18,206)(19,205)(20,204)(21,203)(22,202)(23,201)(24,200)(25,199)(26,198)(27,197)(28,196)(29,210)(30,209)(31,212)(32,211)(33,225)(34,224)(35,223)(36,222)(37,221)(38,220)(39,219)(40,218)(41,217)(42,216)(43,215)(44,214)(45,213)(46,235)(47,234)(48,233)(49,232)(50,231)(51,230)(52,229)(53,228)(54,227)(55,226)(56,240)(57,239)(58,238)(59,237)(60,236)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,135)(68,134)(69,133)(70,132)(71,131)(72,130)(73,129)(74,128)(75,127)(76,140)(77,139)(78,138)(79,137)(80,136)(81,150)(82,149)(83,148)(84,147)(85,146)(86,145)(87,144)(88,143)(89,142)(90,141)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,165)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)(113,180)(114,179)(115,178)(116,177)(117,176)(118,175)(119,174)(120,173) );
G=PermutationGroup([(1,136),(2,137),(3,138),(4,139),(5,140),(6,141),(7,142),(8,143),(9,144),(10,145),(11,146),(12,147),(13,148),(14,149),(15,150),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,133),(23,134),(24,135),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,166),(42,167),(43,168),(44,169),(45,170),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,161),(53,162),(54,163),(55,164),(56,165),(57,151),(58,152),(59,153),(60,154),(61,209),(62,210),(63,196),(64,197),(65,198),(66,199),(67,200),(68,201),(69,202),(70,203),(71,204),(72,205),(73,206),(74,207),(75,208),(76,194),(77,195),(78,181),(79,182),(80,183),(81,184),(82,185),(83,186),(84,187),(85,188),(86,189),(87,190),(88,191),(89,192),(90,193),(91,226),(92,227),(93,228),(94,229),(95,230),(96,231),(97,232),(98,233),(99,234),(100,235),(101,236),(102,237),(103,238),(104,239),(105,240),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217),(113,218),(114,219),(115,220),(116,221),(117,222),(118,223),(119,224),(120,225)], [(1,81,18,74),(2,82,19,75),(3,83,20,61),(4,84,21,62),(5,85,22,63),(6,86,23,64),(7,87,24,65),(8,88,25,66),(9,89,26,67),(10,90,27,68),(11,76,28,69),(12,77,29,70),(13,78,30,71),(14,79,16,72),(15,80,17,73),(31,118,51,91),(32,119,52,92),(33,120,53,93),(34,106,54,94),(35,107,55,95),(36,108,56,96),(37,109,57,97),(38,110,58,98),(39,111,59,99),(40,112,60,100),(41,113,46,101),(42,114,47,102),(43,115,48,103),(44,116,49,104),(45,117,50,105),(121,199,143,191),(122,200,144,192),(123,201,145,193),(124,202,146,194),(125,203,147,195),(126,204,148,181),(127,205,149,182),(128,206,150,183),(129,207,136,184),(130,208,137,185),(131,209,138,186),(132,210,139,187),(133,196,140,188),(134,197,141,189),(135,198,142,190),(151,232,177,214),(152,233,178,215),(153,234,179,216),(154,235,180,217),(155,236,166,218),(156,237,167,219),(157,238,168,220),(158,239,169,221),(159,240,170,222),(160,226,171,223),(161,227,172,224),(162,228,173,225),(163,229,174,211),(164,230,175,212),(165,231,176,213)], [(1,236,18,218),(2,237,19,219),(3,238,20,220),(4,239,21,221),(5,240,22,222),(6,226,23,223),(7,227,24,224),(8,228,25,225),(9,229,26,211),(10,230,27,212),(11,231,28,213),(12,232,29,214),(13,233,30,215),(14,234,16,216),(15,235,17,217),(31,189,51,197),(32,190,52,198),(33,191,53,199),(34,192,54,200),(35,193,55,201),(36,194,56,202),(37,195,57,203),(38,181,58,204),(39,182,59,205),(40,183,60,206),(41,184,46,207),(42,185,47,208),(43,186,48,209),(44,187,49,210),(45,188,50,196),(61,168,83,157),(62,169,84,158),(63,170,85,159),(64,171,86,160),(65,172,87,161),(66,173,88,162),(67,174,89,163),(68,175,90,164),(69,176,76,165),(70,177,77,151),(71,178,78,152),(72,179,79,153),(73,180,80,154),(74,166,81,155),(75,167,82,156),(91,134,118,141),(92,135,119,142),(93,121,120,143),(94,122,106,144),(95,123,107,145),(96,124,108,146),(97,125,109,147),(98,126,110,148),(99,127,111,149),(100,128,112,150),(101,129,113,136),(102,130,114,137),(103,131,115,138),(104,132,116,139),(105,133,117,140)], [(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)], [(1,183),(2,182),(3,181),(4,195),(5,194),(6,193),(7,192),(8,191),(9,190),(10,189),(11,188),(12,187),(13,186),(14,185),(15,184),(16,208),(17,207),(18,206),(19,205),(20,204),(21,203),(22,202),(23,201),(24,200),(25,199),(26,198),(27,197),(28,196),(29,210),(30,209),(31,212),(32,211),(33,225),(34,224),(35,223),(36,222),(37,221),(38,220),(39,219),(40,218),(41,217),(42,216),(43,215),(44,214),(45,213),(46,235),(47,234),(48,233),(49,232),(50,231),(51,230),(52,229),(53,228),(54,227),(55,226),(56,240),(57,239),(58,238),(59,237),(60,236),(61,126),(62,125),(63,124),(64,123),(65,122),(66,121),(67,135),(68,134),(69,133),(70,132),(71,131),(72,130),(73,129),(74,128),(75,127),(76,140),(77,139),(78,138),(79,137),(80,136),(81,150),(82,149),(83,148),(84,147),(85,146),(86,145),(87,144),(88,143),(89,142),(90,141),(91,164),(92,163),(93,162),(94,161),(95,160),(96,159),(97,158),(98,157),(99,156),(100,155),(101,154),(102,153),(103,152),(104,151),(105,165),(106,172),(107,171),(108,170),(109,169),(110,168),(111,167),(112,166),(113,180),(114,179),(115,178),(116,177),(117,176),(118,175),(119,174),(120,173)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 30 |
| 0 | 0 | 0 | 0 | 9 | 41 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 25 |
| 0 | 0 | 0 | 0 | 55 | 24 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 38 | 0 | 0 |
| 0 | 0 | 23 | 53 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 43 | 43 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 56 | 0 | 0 |
| 0 | 0 | 8 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 30 |
| 0 | 0 | 0 | 0 | 5 | 41 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,20,9,0,0,0,0,30,41],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,55,0,0,0,0,25,24],[18,43,0,0,0,0,18,60,0,0,0,0,0,0,5,23,0,0,0,0,38,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[43,1,0,0,0,0,43,18,0,0,0,0,0,0,23,8,0,0,0,0,56,38,0,0,0,0,0,0,20,5,0,0,0,0,30,41] >;
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | ··· | 30 | 2 | 2 | ··· | 2 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D15 | D30 | D30 | Q8⋊3S3 | Q8⋊2D5 | Q8⋊3D15 |
| kernel | C2×Q8⋊3D15 | C2×C4×D15 | C2×D60 | Q8⋊3D15 | Q8×C30 | Q8×C10 | C6×Q8 | C2×C20 | C5×Q8 | C30 | C2×C12 | C3×Q8 | C2×Q8 | C2×C4 | Q8 | C10 | C6 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 4 | 12 | 16 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_3D_{15} % in TeX
G:=Group("C2xQ8:3D15"); // GroupNames label
G:=SmallGroup(480,1173);
// by ID
G=gap.SmallGroup(480,1173);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^15=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀