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## G = C5×Q8.15D6order 480 = 25·3·5

### Direct product of C5 and Q8.15D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×Q8.15D6
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C20 — C5×S3×Q8 — C5×Q8.15D6
 Lower central C3 — C6 — C5×Q8.15D6
 Upper central C1 — C10 — Q8×C10

Generators and relations for C5×Q8.15D6
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 >

Subgroups: 548 in 292 conjugacy classes, 170 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×Q8, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C5×S3, C30, C30, 2- 1+4, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C4○D12, S3×Q8, Q83S3, C6×Q8, C5×Dic3, C60, S3×C10, C2×C30, Q8×C10, Q8×C10, C5×C4○D4, Q8.15D6, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, Q8×C15, C5×2- 1+4, C5×C4○D12, C5×S3×Q8, C5×Q83S3, Q8×C30, C5×Q8.15D6
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C24, C2×C10, C22×S3, C5×S3, 2- 1+4, C22×C10, S3×C23, S3×C10, C23×C10, Q8.15D6, S3×C2×C10, C5×2- 1+4, S3×C22×C10, C5×Q8.15D6

Smallest permutation representation of C5×Q8.15D6
On 240 points
Generators in S240
(1 61 19 59 43)(2 62 20 60 44)(3 63 21 49 45)(4 64 22 50 46)(5 65 23 51 47)(6 66 24 52 48)(7 67 13 53 37)(8 68 14 54 38)(9 69 15 55 39)(10 70 16 56 40)(11 71 17 57 41)(12 72 18 58 42)(25 226 229 203 206)(26 227 230 204 207)(27 228 231 193 208)(28 217 232 194 209)(29 218 233 195 210)(30 219 234 196 211)(31 220 235 197 212)(32 221 236 198 213)(33 222 237 199 214)(34 223 238 200 215)(35 224 239 201 216)(36 225 240 202 205)(73 88 126 107 110)(74 89 127 108 111)(75 90 128 97 112)(76 91 129 98 113)(77 92 130 99 114)(78 93 131 100 115)(79 94 132 101 116)(80 95 121 102 117)(81 96 122 103 118)(82 85 123 104 119)(83 86 124 105 120)(84 87 125 106 109)(133 154 187 165 174)(134 155 188 166 175)(135 156 189 167 176)(136 145 190 168 177)(137 146 191 157 178)(138 147 192 158 179)(139 148 181 159 180)(140 149 182 160 169)(141 150 183 161 170)(142 151 184 162 171)(143 152 185 163 172)(144 153 186 164 173)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 82 79 76)(74 83 80 77)(75 84 81 78)(85 94 91 88)(86 95 92 89)(87 96 93 90)(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)(169 172 175 178)(170 173 176 179)(171 174 177 180)(181 184 187 190)(182 185 188 191)(183 186 189 192)(193 196 199 202)(194 197 200 203)(195 198 201 204)(205 208 211 214)(206 209 212 215)(207 210 213 216)(217 220 223 226)(218 221 224 227)(219 222 225 228)(229 232 235 238)(230 233 236 239)(231 234 237 240)
(1 85 7 91)(2 92 8 86)(3 87 9 93)(4 94 10 88)(5 89 11 95)(6 96 12 90)(13 98 19 104)(14 105 20 99)(15 100 21 106)(16 107 22 101)(17 102 23 108)(18 97 24 103)(25 188 31 182)(26 183 32 189)(27 190 33 184)(28 185 34 191)(29 192 35 186)(30 187 36 181)(37 76 43 82)(38 83 44 77)(39 78 45 84)(40 73 46 79)(41 80 47 74)(42 75 48 81)(49 109 55 115)(50 116 56 110)(51 111 57 117)(52 118 58 112)(53 113 59 119)(54 120 60 114)(61 123 67 129)(62 130 68 124)(63 125 69 131)(64 132 70 126)(65 127 71 121)(66 122 72 128)(133 202 139 196)(134 197 140 203)(135 204 141 198)(136 199 142 193)(137 194 143 200)(138 201 144 195)(145 214 151 208)(146 209 152 215)(147 216 153 210)(148 211 154 205)(149 206 155 212)(150 213 156 207)(157 217 163 223)(158 224 164 218)(159 219 165 225)(160 226 166 220)(161 221 167 227)(162 228 168 222)(169 229 175 235)(170 236 176 230)(171 231 177 237)(172 238 178 232)(173 233 179 239)(174 240 180 234)
(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)
(1 148 7 154)(2 153 8 147)(3 146 9 152)(4 151 10 145)(5 156 11 150)(6 149 12 155)(13 165 19 159)(14 158 20 164)(15 163 21 157)(16 168 22 162)(17 161 23 167)(18 166 24 160)(25 122 31 128)(26 127 32 121)(27 132 33 126)(28 125 34 131)(29 130 35 124)(30 123 36 129)(37 133 43 139)(38 138 44 144)(39 143 45 137)(40 136 46 142)(41 141 47 135)(42 134 48 140)(49 178 55 172)(50 171 56 177)(51 176 57 170)(52 169 58 175)(53 174 59 180)(54 179 60 173)(61 181 67 187)(62 186 68 192)(63 191 69 185)(64 184 70 190)(65 189 71 183)(66 182 72 188)(73 193 79 199)(74 198 80 204)(75 203 81 197)(76 196 82 202)(77 201 83 195)(78 194 84 200)(85 205 91 211)(86 210 92 216)(87 215 93 209)(88 208 94 214)(89 213 95 207)(90 206 96 212)(97 226 103 220)(98 219 104 225)(99 224 105 218)(100 217 106 223)(101 222 107 228)(102 227 108 221)(109 238 115 232)(110 231 116 237)(111 236 117 230)(112 229 118 235)(113 234 119 240)(114 239 120 233)

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

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

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

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

Matrix representation of C5×Q8.15D6 in 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] >;

C5×Q8.15D6 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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