direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8.15D6, C30.93C24, C60.240C23, C15⋊102- (1+4), (S3×Q8)⋊4C10, (C6×Q8)⋊7C10, C4○D12⋊6C10, (Q8×C10)⋊16S3, (Q8×C30)⋊21C2, (C5×Q8).59D6, Q8⋊3S3⋊4C10, (C2×C20).253D6, Q8.15(S3×C10), D12.13(C2×C10), C3⋊1(C5×2- (1+4)), C10.78(S3×C23), C6.10(C23×C10), D6.5(C22×C10), (S3×C10).40C23, (S3×C20).40C22, (C2×C30).448C23, C12.24(C22×C10), (C2×C60).376C22, C20.213(C22×S3), Dic6.13(C2×C10), (C5×D12).52C22, (Q8×C15).53C22, (C5×Dic3).42C23, Dic3.6(C22×C10), (C5×Dic6).55C22, (C5×S3×Q8)⋊11C2, C4.24(S3×C2×C10), (C2×Q8)⋊7(C5×S3), C22.7(S3×C2×C10), (C5×C4○D12)⋊16C2, (C4×S3).5(C2×C10), (C2×C4).22(S3×C10), C3⋊D4.2(C2×C10), C2.11(S3×C22×C10), (C2×C12).50(C2×C10), (C5×Q8⋊3S3)⋊11C2, (C3×Q8).10(C2×C10), (C5×C3⋊D4).5C22, (C2×C6).68(C22×C10), (C2×C10).259(C22×S3), SmallGroup(480,1159)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 548 in 292 conjugacy classes, 170 normal (18 characteristic)
C1, C2, C2 [×5], C3, C4 [×6], C4 [×4], C22, C22 [×4], C5, S3 [×4], C6, C6, C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], C10, C10 [×5], Dic3 [×4], C12 [×6], D6 [×4], C2×C6, C15, C2×Q8, C2×Q8 [×4], C4○D4 [×10], C20 [×6], C20 [×4], C2×C10, C2×C10 [×4], Dic6 [×6], C4×S3 [×12], D12 [×6], C3⋊D4 [×4], C2×C12 [×3], C3×Q8 [×4], C5×S3 [×4], C30, C30, 2- (1+4), C2×C20 [×3], C2×C20 [×12], C5×D4 [×10], C5×Q8 [×4], C5×Q8 [×6], C4○D12 [×6], S3×Q8 [×4], Q8⋊3S3 [×4], C6×Q8, C5×Dic3 [×4], C60 [×6], S3×C10 [×4], C2×C30, Q8×C10, Q8×C10 [×4], C5×C4○D4 [×10], Q8.15D6, C5×Dic6 [×6], S3×C20 [×12], C5×D12 [×6], C5×C3⋊D4 [×4], C2×C60 [×3], Q8×C15 [×4], C5×2- (1+4), C5×C4○D12 [×6], C5×S3×Q8 [×4], C5×Q8⋊3S3 [×4], Q8×C30, C5×Q8.15D6
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, 2- (1+4), C22×C10 [×15], S3×C23, S3×C10 [×7], C23×C10, Q8.15D6, S3×C2×C10 [×7], C5×2- (1+4), S3×C22×C10, C5×Q8.15D6
Generators and relations
G = < a,b,c,d,e | a5=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d5 >
►On 240 points
Generators in S240
(1 223 176 232 100)(2 224 177 233 101)(3 225 178 234 102)(4 226 179 235 103)(5 227 180 236 104)(6 228 169 237 105)(7 217 170 238 106)(8 218 171 239 107)(9 219 172 240 108)(10 220 173 229 97)(11 221 174 230 98)(12 222 175 231 99)(13 78 96 142 210)(14 79 85 143 211)(15 80 86 144 212)(16 81 87 133 213)(17 82 88 134 214)(18 83 89 135 215)(19 84 90 136 216)(20 73 91 137 205)(21 74 92 138 206)(22 75 93 139 207)(23 76 94 140 208)(24 77 95 141 209)(25 68 42 166 154)(26 69 43 167 155)(27 70 44 168 156)(28 71 45 157 145)(29 72 46 158 146)(30 61 47 159 147)(31 62 48 160 148)(32 63 37 161 149)(33 64 38 162 150)(34 65 39 163 151)(35 66 40 164 152)(36 67 41 165 153)(49 202 110 185 126)(50 203 111 186 127)(51 204 112 187 128)(52 193 113 188 129)(53 194 114 189 130)(54 195 115 190 131)(55 196 116 191 132)(56 197 117 192 121)(57 198 118 181 122)(58 199 119 182 123)(59 200 120 183 124)(60 201 109 184 125)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 106 103 100)(98 107 104 101)(99 108 105 102)(109 118 115 112)(110 119 116 113)(111 120 117 114)(121 130 127 124)(122 131 128 125)(123 132 129 126)(133 136 139 142)(134 137 140 143)(135 138 141 144)(145 148 151 154)(146 149 152 155)(147 150 153 156)(157 160 163 166)(158 161 164 167)(159 162 165 168)(169 178 175 172)(170 179 176 173)(171 180 177 174)(181 190 187 184)(182 191 188 185)(183 192 189 186)(193 202 199 196)(194 203 200 197)(195 204 201 198)(205 208 211 214)(206 209 212 215)(207 210 213 216)(217 226 223 220)(218 227 224 221)(219 228 225 222)(229 238 235 232)(230 239 236 233)(231 240 237 234)
(1 122 7 128)(2 129 8 123)(3 124 9 130)(4 131 10 125)(5 126 11 132)(6 121 12 127)(13 31 19 25)(14 26 20 32)(15 33 21 27)(16 28 22 34)(17 35 23 29)(18 30 24 36)(37 85 43 91)(38 92 44 86)(39 87 45 93)(40 94 46 88)(41 89 47 95)(42 96 48 90)(49 221 55 227)(50 228 56 222)(51 223 57 217)(52 218 58 224)(53 225 59 219)(54 220 60 226)(61 77 67 83)(62 84 68 78)(63 79 69 73)(64 74 70 80)(65 81 71 75)(66 76 72 82)(97 184 103 190)(98 191 104 185)(99 186 105 192)(100 181 106 187)(101 188 107 182)(102 183 108 189)(109 235 115 229)(110 230 116 236)(111 237 117 231)(112 232 118 238)(113 239 119 233)(114 234 120 240)(133 157 139 163)(134 164 140 158)(135 159 141 165)(136 166 142 160)(137 161 143 167)(138 168 144 162)(145 207 151 213)(146 214 152 208)(147 209 153 215)(148 216 154 210)(149 211 155 205)(150 206 156 212)(169 197 175 203)(170 204 176 198)(171 199 177 193)(172 194 178 200)(173 201 179 195)(174 196 180 202)
(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)
(1 206 7 212)(2 211 8 205)(3 216 9 210)(4 209 10 215)(5 214 11 208)(6 207 12 213)(13 225 19 219)(14 218 20 224)(15 223 21 217)(16 228 22 222)(17 221 23 227)(18 226 24 220)(25 59 31 53)(26 52 32 58)(27 57 33 51)(28 50 34 56)(29 55 35 49)(30 60 36 54)(37 119 43 113)(38 112 44 118)(39 117 45 111)(40 110 46 116)(41 115 47 109)(42 120 48 114)(61 201 67 195)(62 194 68 200)(63 199 69 193)(64 204 70 198)(65 197 71 203)(66 202 72 196)(73 177 79 171)(74 170 80 176)(75 175 81 169)(76 180 82 174)(77 173 83 179)(78 178 84 172)(85 239 91 233)(86 232 92 238)(87 237 93 231)(88 230 94 236)(89 235 95 229)(90 240 96 234)(97 135 103 141)(98 140 104 134)(99 133 105 139)(100 138 106 144)(101 143 107 137)(102 136 108 142)(121 145 127 151)(122 150 128 156)(123 155 129 149)(124 148 130 154)(125 153 131 147)(126 146 132 152)(157 186 163 192)(158 191 164 185)(159 184 165 190)(160 189 166 183)(161 182 167 188)(162 187 168 181)
G:=sub<Sym(240)| (1,223,176,232,100)(2,224,177,233,101)(3,225,178,234,102)(4,226,179,235,103)(5,227,180,236,104)(6,228,169,237,105)(7,217,170,238,106)(8,218,171,239,107)(9,219,172,240,108)(10,220,173,229,97)(11,221,174,230,98)(12,222,175,231,99)(13,78,96,142,210)(14,79,85,143,211)(15,80,86,144,212)(16,81,87,133,213)(17,82,88,134,214)(18,83,89,135,215)(19,84,90,136,216)(20,73,91,137,205)(21,74,92,138,206)(22,75,93,139,207)(23,76,94,140,208)(24,77,95,141,209)(25,68,42,166,154)(26,69,43,167,155)(27,70,44,168,156)(28,71,45,157,145)(29,72,46,158,146)(30,61,47,159,147)(31,62,48,160,148)(32,63,37,161,149)(33,64,38,162,150)(34,65,39,163,151)(35,66,40,164,152)(36,67,41,165,153)(49,202,110,185,126)(50,203,111,186,127)(51,204,112,187,128)(52,193,113,188,129)(53,194,114,189,130)(54,195,115,190,131)(55,196,116,191,132)(56,197,117,192,121)(57,198,118,181,122)(58,199,119,182,123)(59,200,120,183,124)(60,201,109,184,125), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,106,103,100)(98,107,104,101)(99,108,105,102)(109,118,115,112)(110,119,116,113)(111,120,117,114)(121,130,127,124)(122,131,128,125)(123,132,129,126)(133,136,139,142)(134,137,140,143)(135,138,141,144)(145,148,151,154)(146,149,152,155)(147,150,153,156)(157,160,163,166)(158,161,164,167)(159,162,165,168)(169,178,175,172)(170,179,176,173)(171,180,177,174)(181,190,187,184)(182,191,188,185)(183,192,189,186)(193,202,199,196)(194,203,200,197)(195,204,201,198)(205,208,211,214)(206,209,212,215)(207,210,213,216)(217,226,223,220)(218,227,224,221)(219,228,225,222)(229,238,235,232)(230,239,236,233)(231,240,237,234), (1,122,7,128)(2,129,8,123)(3,124,9,130)(4,131,10,125)(5,126,11,132)(6,121,12,127)(13,31,19,25)(14,26,20,32)(15,33,21,27)(16,28,22,34)(17,35,23,29)(18,30,24,36)(37,85,43,91)(38,92,44,86)(39,87,45,93)(40,94,46,88)(41,89,47,95)(42,96,48,90)(49,221,55,227)(50,228,56,222)(51,223,57,217)(52,218,58,224)(53,225,59,219)(54,220,60,226)(61,77,67,83)(62,84,68,78)(63,79,69,73)(64,74,70,80)(65,81,71,75)(66,76,72,82)(97,184,103,190)(98,191,104,185)(99,186,105,192)(100,181,106,187)(101,188,107,182)(102,183,108,189)(109,235,115,229)(110,230,116,236)(111,237,117,231)(112,232,118,238)(113,239,119,233)(114,234,120,240)(133,157,139,163)(134,164,140,158)(135,159,141,165)(136,166,142,160)(137,161,143,167)(138,168,144,162)(145,207,151,213)(146,214,152,208)(147,209,153,215)(148,216,154,210)(149,211,155,205)(150,206,156,212)(169,197,175,203)(170,204,176,198)(171,199,177,193)(172,194,178,200)(173,201,179,195)(174,196,180,202), (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), (1,206,7,212)(2,211,8,205)(3,216,9,210)(4,209,10,215)(5,214,11,208)(6,207,12,213)(13,225,19,219)(14,218,20,224)(15,223,21,217)(16,228,22,222)(17,221,23,227)(18,226,24,220)(25,59,31,53)(26,52,32,58)(27,57,33,51)(28,50,34,56)(29,55,35,49)(30,60,36,54)(37,119,43,113)(38,112,44,118)(39,117,45,111)(40,110,46,116)(41,115,47,109)(42,120,48,114)(61,201,67,195)(62,194,68,200)(63,199,69,193)(64,204,70,198)(65,197,71,203)(66,202,72,196)(73,177,79,171)(74,170,80,176)(75,175,81,169)(76,180,82,174)(77,173,83,179)(78,178,84,172)(85,239,91,233)(86,232,92,238)(87,237,93,231)(88,230,94,236)(89,235,95,229)(90,240,96,234)(97,135,103,141)(98,140,104,134)(99,133,105,139)(100,138,106,144)(101,143,107,137)(102,136,108,142)(121,145,127,151)(122,150,128,156)(123,155,129,149)(124,148,130,154)(125,153,131,147)(126,146,132,152)(157,186,163,192)(158,191,164,185)(159,184,165,190)(160,189,166,183)(161,182,167,188)(162,187,168,181)>;
G:=Group( (1,223,176,232,100)(2,224,177,233,101)(3,225,178,234,102)(4,226,179,235,103)(5,227,180,236,104)(6,228,169,237,105)(7,217,170,238,106)(8,218,171,239,107)(9,219,172,240,108)(10,220,173,229,97)(11,221,174,230,98)(12,222,175,231,99)(13,78,96,142,210)(14,79,85,143,211)(15,80,86,144,212)(16,81,87,133,213)(17,82,88,134,214)(18,83,89,135,215)(19,84,90,136,216)(20,73,91,137,205)(21,74,92,138,206)(22,75,93,139,207)(23,76,94,140,208)(24,77,95,141,209)(25,68,42,166,154)(26,69,43,167,155)(27,70,44,168,156)(28,71,45,157,145)(29,72,46,158,146)(30,61,47,159,147)(31,62,48,160,148)(32,63,37,161,149)(33,64,38,162,150)(34,65,39,163,151)(35,66,40,164,152)(36,67,41,165,153)(49,202,110,185,126)(50,203,111,186,127)(51,204,112,187,128)(52,193,113,188,129)(53,194,114,189,130)(54,195,115,190,131)(55,196,116,191,132)(56,197,117,192,121)(57,198,118,181,122)(58,199,119,182,123)(59,200,120,183,124)(60,201,109,184,125), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,106,103,100)(98,107,104,101)(99,108,105,102)(109,118,115,112)(110,119,116,113)(111,120,117,114)(121,130,127,124)(122,131,128,125)(123,132,129,126)(133,136,139,142)(134,137,140,143)(135,138,141,144)(145,148,151,154)(146,149,152,155)(147,150,153,156)(157,160,163,166)(158,161,164,167)(159,162,165,168)(169,178,175,172)(170,179,176,173)(171,180,177,174)(181,190,187,184)(182,191,188,185)(183,192,189,186)(193,202,199,196)(194,203,200,197)(195,204,201,198)(205,208,211,214)(206,209,212,215)(207,210,213,216)(217,226,223,220)(218,227,224,221)(219,228,225,222)(229,238,235,232)(230,239,236,233)(231,240,237,234), (1,122,7,128)(2,129,8,123)(3,124,9,130)(4,131,10,125)(5,126,11,132)(6,121,12,127)(13,31,19,25)(14,26,20,32)(15,33,21,27)(16,28,22,34)(17,35,23,29)(18,30,24,36)(37,85,43,91)(38,92,44,86)(39,87,45,93)(40,94,46,88)(41,89,47,95)(42,96,48,90)(49,221,55,227)(50,228,56,222)(51,223,57,217)(52,218,58,224)(53,225,59,219)(54,220,60,226)(61,77,67,83)(62,84,68,78)(63,79,69,73)(64,74,70,80)(65,81,71,75)(66,76,72,82)(97,184,103,190)(98,191,104,185)(99,186,105,192)(100,181,106,187)(101,188,107,182)(102,183,108,189)(109,235,115,229)(110,230,116,236)(111,237,117,231)(112,232,118,238)(113,239,119,233)(114,234,120,240)(133,157,139,163)(134,164,140,158)(135,159,141,165)(136,166,142,160)(137,161,143,167)(138,168,144,162)(145,207,151,213)(146,214,152,208)(147,209,153,215)(148,216,154,210)(149,211,155,205)(150,206,156,212)(169,197,175,203)(170,204,176,198)(171,199,177,193)(172,194,178,200)(173,201,179,195)(174,196,180,202), (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), (1,206,7,212)(2,211,8,205)(3,216,9,210)(4,209,10,215)(5,214,11,208)(6,207,12,213)(13,225,19,219)(14,218,20,224)(15,223,21,217)(16,228,22,222)(17,221,23,227)(18,226,24,220)(25,59,31,53)(26,52,32,58)(27,57,33,51)(28,50,34,56)(29,55,35,49)(30,60,36,54)(37,119,43,113)(38,112,44,118)(39,117,45,111)(40,110,46,116)(41,115,47,109)(42,120,48,114)(61,201,67,195)(62,194,68,200)(63,199,69,193)(64,204,70,198)(65,197,71,203)(66,202,72,196)(73,177,79,171)(74,170,80,176)(75,175,81,169)(76,180,82,174)(77,173,83,179)(78,178,84,172)(85,239,91,233)(86,232,92,238)(87,237,93,231)(88,230,94,236)(89,235,95,229)(90,240,96,234)(97,135,103,141)(98,140,104,134)(99,133,105,139)(100,138,106,144)(101,143,107,137)(102,136,108,142)(121,145,127,151)(122,150,128,156)(123,155,129,149)(124,148,130,154)(125,153,131,147)(126,146,132,152)(157,186,163,192)(158,191,164,185)(159,184,165,190)(160,189,166,183)(161,182,167,188)(162,187,168,181) );
G=PermutationGroup([(1,223,176,232,100),(2,224,177,233,101),(3,225,178,234,102),(4,226,179,235,103),(5,227,180,236,104),(6,228,169,237,105),(7,217,170,238,106),(8,218,171,239,107),(9,219,172,240,108),(10,220,173,229,97),(11,221,174,230,98),(12,222,175,231,99),(13,78,96,142,210),(14,79,85,143,211),(15,80,86,144,212),(16,81,87,133,213),(17,82,88,134,214),(18,83,89,135,215),(19,84,90,136,216),(20,73,91,137,205),(21,74,92,138,206),(22,75,93,139,207),(23,76,94,140,208),(24,77,95,141,209),(25,68,42,166,154),(26,69,43,167,155),(27,70,44,168,156),(28,71,45,157,145),(29,72,46,158,146),(30,61,47,159,147),(31,62,48,160,148),(32,63,37,161,149),(33,64,38,162,150),(34,65,39,163,151),(35,66,40,164,152),(36,67,41,165,153),(49,202,110,185,126),(50,203,111,186,127),(51,204,112,187,128),(52,193,113,188,129),(53,194,114,189,130),(54,195,115,190,131),(55,196,116,191,132),(56,197,117,192,121),(57,198,118,181,122),(58,199,119,182,123),(59,200,120,183,124),(60,201,109,184,125)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,58,55,52),(50,59,56,53),(51,60,57,54),(61,64,67,70),(62,65,68,71),(63,66,69,72),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,88,91,94),(86,89,92,95),(87,90,93,96),(97,106,103,100),(98,107,104,101),(99,108,105,102),(109,118,115,112),(110,119,116,113),(111,120,117,114),(121,130,127,124),(122,131,128,125),(123,132,129,126),(133,136,139,142),(134,137,140,143),(135,138,141,144),(145,148,151,154),(146,149,152,155),(147,150,153,156),(157,160,163,166),(158,161,164,167),(159,162,165,168),(169,178,175,172),(170,179,176,173),(171,180,177,174),(181,190,187,184),(182,191,188,185),(183,192,189,186),(193,202,199,196),(194,203,200,197),(195,204,201,198),(205,208,211,214),(206,209,212,215),(207,210,213,216),(217,226,223,220),(218,227,224,221),(219,228,225,222),(229,238,235,232),(230,239,236,233),(231,240,237,234)], [(1,122,7,128),(2,129,8,123),(3,124,9,130),(4,131,10,125),(5,126,11,132),(6,121,12,127),(13,31,19,25),(14,26,20,32),(15,33,21,27),(16,28,22,34),(17,35,23,29),(18,30,24,36),(37,85,43,91),(38,92,44,86),(39,87,45,93),(40,94,46,88),(41,89,47,95),(42,96,48,90),(49,221,55,227),(50,228,56,222),(51,223,57,217),(52,218,58,224),(53,225,59,219),(54,220,60,226),(61,77,67,83),(62,84,68,78),(63,79,69,73),(64,74,70,80),(65,81,71,75),(66,76,72,82),(97,184,103,190),(98,191,104,185),(99,186,105,192),(100,181,106,187),(101,188,107,182),(102,183,108,189),(109,235,115,229),(110,230,116,236),(111,237,117,231),(112,232,118,238),(113,239,119,233),(114,234,120,240),(133,157,139,163),(134,164,140,158),(135,159,141,165),(136,166,142,160),(137,161,143,167),(138,168,144,162),(145,207,151,213),(146,214,152,208),(147,209,153,215),(148,216,154,210),(149,211,155,205),(150,206,156,212),(169,197,175,203),(170,204,176,198),(171,199,177,193),(172,194,178,200),(173,201,179,195),(174,196,180,202)], [(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)], [(1,206,7,212),(2,211,8,205),(3,216,9,210),(4,209,10,215),(5,214,11,208),(6,207,12,213),(13,225,19,219),(14,218,20,224),(15,223,21,217),(16,228,22,222),(17,221,23,227),(18,226,24,220),(25,59,31,53),(26,52,32,58),(27,57,33,51),(28,50,34,56),(29,55,35,49),(30,60,36,54),(37,119,43,113),(38,112,44,118),(39,117,45,111),(40,110,46,116),(41,115,47,109),(42,120,48,114),(61,201,67,195),(62,194,68,200),(63,199,69,193),(64,204,70,198),(65,197,71,203),(66,202,72,196),(73,177,79,171),(74,170,80,176),(75,175,81,169),(76,180,82,174),(77,173,83,179),(78,178,84,172),(85,239,91,233),(86,232,92,238),(87,237,93,231),(88,230,94,236),(89,235,95,229),(90,240,96,234),(97,135,103,141),(98,140,104,134),(99,133,105,139),(100,138,106,144),(101,143,107,137),(102,136,108,142),(121,145,127,151),(122,150,128,156),(123,155,129,149),(124,148,130,154),(125,153,131,147),(126,146,132,152),(157,186,163,192),(158,191,164,185),(159,184,165,190),(160,189,166,183),(161,182,167,188),(162,187,168,181)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 58 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 60 | 15 | 24 | 60 |
| 8 | 1 | 22 | 30 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 |
| 48 | 8 | 16 | 41 |
| 55 | 13 | 2 | 8 |
| 0 | 0 | 32 | 16 |
| 0 | 0 | 16 | 29 |
| 48 | 12 | 19 | 35 |
| 43 | 13 | 42 | 28 |
| 0 | 0 | 0 | 47 |
| 0 | 0 | 14 | 0 |
| 7 | 5 | 44 | 45 |
| 0 | 48 | 4 | 0 |
| 0 | 49 | 13 | 0 |
| 26 | 49 | 41 | 54 |
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[60,8,0,0,15,1,0,0,24,22,0,1,60,30,60,0],[48,55,0,0,8,13,0,0,16,2,32,16,41,8,16,29],[48,43,0,0,12,13,0,0,19,42,0,14,35,28,47,0],[7,0,0,26,5,48,49,49,44,4,13,41,45,0,0,54] >;
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10X | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 20Y | ··· | 20AN | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | C5×S3 | S3×C10 | S3×C10 | 2- (1+4) | Q8.15D6 | C5×2- (1+4) | C5×Q8.15D6 |
| kernel | C5×Q8.15D6 | C5×C4○D12 | C5×S3×Q8 | C5×Q8⋊3S3 | Q8×C30 | Q8.15D6 | C4○D12 | S3×Q8 | Q8⋊3S3 | C6×Q8 | Q8×C10 | C2×C20 | C5×Q8 | C2×Q8 | C2×C4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 4 | 24 | 16 | 16 | 4 | 1 | 3 | 4 | 4 | 12 | 16 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_5\times Q_8._{15}D_6 % in TeX
G:=Group("C5xQ8.15D6"); // GroupNames label
G:=SmallGroup(480,1159);
// by ID
G=gap.SmallGroup(480,1159);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,568,891,436,2467,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=1,c^2=d^6=e^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^-1=b^-1,b*d=d*b,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^5>;
// generators/relations
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