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G = C5×Q8○D12order 480 = 25·3·5

Direct product of C5 and Q8○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×Q8○D12
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C20 — C5×S3×Q8 — C5×Q8○D12
 Lower central C3 — C6 — C5×Q8○D12
 Upper central C1 — C10 — C5×C4○D4

Generators and relations for C5×Q8○D12
G = < a,b,c,d,e | a5=b4=e2=1, c2=d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 532 in 292 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, S3 [×2], C6, C6 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], C10, C10 [×5], Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C15, C2×Q8 [×5], C4○D4, C4○D4 [×9], C20, C20 [×3], C20 [×6], C2×C10 [×3], C2×C10 [×2], Dic6 [×9], C4×S3 [×6], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C5×S3 [×2], C30, C30 [×3], 2- 1+4, C2×C20 [×3], C2×C20 [×12], C5×D4 [×3], C5×D4 [×7], C5×Q8, C5×Q8 [×9], C2×Dic6 [×3], C4○D12 [×3], D42S3 [×6], S3×Q8 [×2], C3×C4○D4, C5×Dic3 [×6], C60, C60 [×3], S3×C10 [×2], C2×C30 [×3], Q8×C10 [×5], C5×C4○D4, C5×C4○D4 [×9], Q8○D12, C5×Dic6 [×9], S3×C20 [×6], C5×D12, C10×Dic3 [×6], C5×C3⋊D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, C5×2- 1+4, C10×Dic6 [×3], C5×C4○D12 [×3], C5×D42S3 [×6], C5×S3×Q8 [×2], C15×C4○D4, C5×Q8○D12
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○D12, S3×C2×C10 [×7], C5×2- 1+4, S3×C22×C10, C5×Q8○D12

Smallest permutation representation of C5×Q8○D12
On 240 points
Generators in S240
(1 67 16 53 41)(2 68 17 54 42)(3 69 18 55 43)(4 70 19 56 44)(5 71 20 57 45)(6 72 21 58 46)(7 61 22 59 47)(8 62 23 60 48)(9 63 24 49 37)(10 64 13 50 38)(11 65 14 51 39)(12 66 15 52 40)(25 217 232 195 210)(26 218 233 196 211)(27 219 234 197 212)(28 220 235 198 213)(29 221 236 199 214)(30 222 237 200 215)(31 223 238 201 216)(32 224 239 202 205)(33 225 240 203 206)(34 226 229 204 207)(35 227 230 193 208)(36 228 231 194 209)(73 88 124 107 110)(74 89 125 108 111)(75 90 126 97 112)(76 91 127 98 113)(77 92 128 99 114)(78 93 129 100 115)(79 94 130 101 116)(80 95 131 102 117)(81 96 132 103 118)(82 85 121 104 119)(83 86 122 105 120)(84 87 123 106 109)(133 148 184 167 170)(134 149 185 168 171)(135 150 186 157 172)(136 151 187 158 173)(137 152 188 159 174)(138 153 189 160 175)(139 154 190 161 176)(140 155 191 162 177)(141 156 192 163 178)(142 145 181 164 179)(143 146 182 165 180)(144 147 183 166 169)
(1 145 7 151)(2 146 8 152)(3 147 9 153)(4 148 10 154)(5 149 11 155)(6 150 12 156)(13 161 19 167)(14 162 20 168)(15 163 21 157)(16 164 22 158)(17 165 23 159)(18 166 24 160)(25 126 31 132)(26 127 32 121)(27 128 33 122)(28 129 34 123)(29 130 35 124)(30 131 36 125)(37 138 43 144)(38 139 44 133)(39 140 45 134)(40 141 46 135)(41 142 47 136)(42 143 48 137)(49 175 55 169)(50 176 56 170)(51 177 57 171)(52 178 58 172)(53 179 59 173)(54 180 60 174)(61 187 67 181)(62 188 68 182)(63 189 69 183)(64 190 70 184)(65 191 71 185)(66 192 72 186)(73 199 79 193)(74 200 80 194)(75 201 81 195)(76 202 82 196)(77 203 83 197)(78 204 84 198)(85 211 91 205)(86 212 92 206)(87 213 93 207)(88 214 94 208)(89 215 95 209)(90 216 96 210)(97 223 103 217)(98 224 104 218)(99 225 105 219)(100 226 106 220)(101 227 107 221)(102 228 108 222)(109 235 115 229)(110 236 116 230)(111 237 117 231)(112 238 118 232)(113 239 119 233)(114 240 120 234)
(1 211 7 205)(2 212 8 206)(3 213 9 207)(4 214 10 208)(5 215 11 209)(6 216 12 210)(13 227 19 221)(14 228 20 222)(15 217 21 223)(16 218 22 224)(17 219 23 225)(18 220 24 226)(25 72 31 66)(26 61 32 67)(27 62 33 68)(28 63 34 69)(29 64 35 70)(30 65 36 71)(37 204 43 198)(38 193 44 199)(39 194 45 200)(40 195 46 201)(41 196 47 202)(42 197 48 203)(49 229 55 235)(50 230 56 236)(51 231 57 237)(52 232 58 238)(53 233 59 239)(54 234 60 240)(73 139 79 133)(74 140 80 134)(75 141 81 135)(76 142 82 136)(77 143 83 137)(78 144 84 138)(85 151 91 145)(86 152 92 146)(87 153 93 147)(88 154 94 148)(89 155 95 149)(90 156 96 150)(97 163 103 157)(98 164 104 158)(99 165 105 159)(100 166 106 160)(101 167 107 161)(102 168 108 162)(109 175 115 169)(110 176 116 170)(111 177 117 171)(112 178 118 172)(113 179 119 173)(114 180 120 174)(121 187 127 181)(122 188 128 182)(123 189 129 183)(124 190 130 184)(125 191 131 185)(126 192 132 186)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 21)(14 20)(15 19)(16 18)(22 24)(25 29)(26 28)(30 36)(31 35)(32 34)(37 47)(38 46)(39 45)(40 44)(41 43)(49 59)(50 58)(51 57)(52 56)(53 55)(61 63)(64 72)(65 71)(66 70)(67 69)(73 81)(74 80)(75 79)(76 78)(82 84)(85 87)(88 96)(89 95)(90 94)(91 93)(97 101)(98 100)(102 108)(103 107)(104 106)(109 119)(110 118)(111 117)(112 116)(113 115)(121 123)(124 132)(125 131)(126 130)(127 129)(133 141)(134 140)(135 139)(136 138)(142 144)(145 147)(148 156)(149 155)(150 154)(151 153)(157 161)(158 160)(162 168)(163 167)(164 166)(169 179)(170 178)(171 177)(172 176)(173 175)(181 183)(184 192)(185 191)(186 190)(187 189)(193 201)(194 200)(195 199)(196 198)(202 204)(205 207)(208 216)(209 215)(210 214)(211 213)(217 221)(218 220)(222 228)(223 227)(224 226)(229 239)(230 238)(231 237)(232 236)(233 235)

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

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

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

135 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E ··· 4J 5A 5B 5C 5D 6A 6B 6C 6D 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10X 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AN 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 5 5 5 5 6 6 6 6 10 10 10 10 10 ··· 10 10 ··· 10 12 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 2 2 6 6 2 2 2 2 2 6 ··· 6 1 1 1 1 2 4 4 4 1 1 1 1 2 ··· 2 6 ··· 6 2 2 4 4 4 2 2 2 2 2 ··· 2 6 ··· 6 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D6 D6 D6 C5×S3 S3×C10 S3×C10 S3×C10 2- 1+4 Q8○D12 C5×2- 1+4 C5×Q8○D12 kernel C5×Q8○D12 C10×Dic6 C5×C4○D12 C5×D4⋊2S3 C5×S3×Q8 C15×C4○D4 Q8○D12 C2×Dic6 C4○D12 D4⋊2S3 S3×Q8 C3×C4○D4 C5×C4○D4 C2×C20 C5×D4 C5×Q8 C4○D4 C2×C4 D4 Q8 C15 C5 C3 C1 # reps 1 3 3 6 2 1 4 12 12 24 8 4 1 3 3 1 4 12 12 4 1 2 4 8

Matrix representation of C5×Q8○D12 in GL4(𝔽61) generated by

 58 0 0 0 0 58 0 0 0 0 58 0 0 0 0 58
,
 34 0 35 0 0 34 0 35 14 0 27 0 0 14 0 27
,
 1 0 28 0 0 1 0 28 13 0 60 0 0 13 0 60
,
 46 23 0 0 38 23 0 0 0 0 46 23 0 0 38 23
,
 0 60 0 0 60 0 0 0 0 0 0 60 0 0 60 0
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[34,0,14,0,0,34,0,14,35,0,27,0,0,35,0,27],[1,0,13,0,0,1,0,13,28,0,60,0,0,28,0,60],[46,38,0,0,23,23,0,0,0,0,46,38,0,0,23,23],[0,60,0,0,60,0,0,0,0,0,0,60,0,0,60,0] >;

C5×Q8○D12 in GAP, Magma, Sage, TeX

C_5\times Q_8\circ D_{12}
% in TeX

G:=Group("C5xQ8oD12");
// GroupNames label

G:=SmallGroup(480,1162);
// by ID

G=gap.SmallGroup(480,1162);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,568,891,436,2467,304,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=e^2=1,c^2=d^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^5>;
// generators/relations

׿
×
𝔽