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## G = C5×C23.21D6order 480 = 25·3·5

### Direct product of C5 and C23.21D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C23.21D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — C10×C3⋊D4 — C5×C23.21D6
 Lower central C3 — C2×C6 — C5×C23.21D6
 Upper central C1 — C2×C10 — C5×C22⋊C4

Generators and relations for C5×C23.21D6
G = < a,b,c,d,e | a5=b2=c2=d12=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 372 in 156 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C30, C22.D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C23.21D6, C10×Dic3, C10×Dic3, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, C22×C30, C5×C22.D4, C5×C4⋊Dic3, C5×D6⋊C4, C15×C22⋊C4, Dic3×C2×C10, C10×C3⋊D4, C5×C23.21D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, D12, C22×S3, C5×S3, C22.D4, C5×D4, C22×C10, C2×D12, D42S3, S3×C10, D4×C10, C5×C4○D4, C23.21D6, C5×D12, S3×C2×C10, C5×C22.D4, C10×D12, C5×D42S3, C5×C23.21D6

Smallest permutation representation of C5×C23.21D6
On 240 points
Generators in S240
(1 205 173 165 27)(2 206 174 166 28)(3 207 175 167 29)(4 208 176 168 30)(5 209 177 157 31)(6 210 178 158 32)(7 211 179 159 33)(8 212 180 160 34)(9 213 169 161 35)(10 214 170 162 36)(11 215 171 163 25)(12 216 172 164 26)(13 56 41 108 183)(14 57 42 97 184)(15 58 43 98 185)(16 59 44 99 186)(17 60 45 100 187)(18 49 46 101 188)(19 50 47 102 189)(20 51 48 103 190)(21 52 37 104 191)(22 53 38 105 192)(23 54 39 106 181)(24 55 40 107 182)(61 238 155 109 125)(62 239 156 110 126)(63 240 145 111 127)(64 229 146 112 128)(65 230 147 113 129)(66 231 148 114 130)(67 232 149 115 131)(68 233 150 116 132)(69 234 151 117 121)(70 235 152 118 122)(71 236 153 119 123)(72 237 154 120 124)(73 204 91 139 218)(74 193 92 140 219)(75 194 93 141 220)(76 195 94 142 221)(77 196 95 143 222)(78 197 96 144 223)(79 198 85 133 224)(80 199 86 134 225)(81 200 87 135 226)(82 201 88 136 227)(83 202 89 137 228)(84 203 90 138 217)
(2 117)(4 119)(6 109)(8 111)(10 113)(12 115)(14 226)(16 228)(18 218)(20 220)(22 222)(24 224)(26 149)(28 151)(30 153)(32 155)(34 145)(36 147)(38 196)(40 198)(42 200)(44 202)(46 204)(48 194)(49 73)(51 75)(53 77)(55 79)(57 81)(59 83)(61 178)(63 180)(65 170)(67 172)(69 174)(71 176)(85 107)(87 97)(89 99)(91 101)(93 103)(95 105)(121 206)(123 208)(125 210)(127 212)(129 214)(131 216)(133 182)(135 184)(137 186)(139 188)(141 190)(143 192)(158 238)(160 240)(162 230)(164 232)(166 234)(168 236)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 225)(14 226)(15 227)(16 228)(17 217)(18 218)(19 219)(20 220)(21 221)(22 222)(23 223)(24 224)(25 148)(26 149)(27 150)(28 151)(29 152)(30 153)(31 154)(32 155)(33 156)(34 145)(35 146)(36 147)(37 195)(38 196)(39 197)(40 198)(41 199)(42 200)(43 201)(44 202)(45 203)(46 204)(47 193)(48 194)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 178)(62 179)(63 180)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 176)(72 177)(85 107)(86 108)(87 97)(88 98)(89 99)(90 100)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(121 206)(122 207)(123 208)(124 209)(125 210)(126 211)(127 212)(128 213)(129 214)(130 215)(131 216)(132 205)(133 182)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 190)(142 191)(143 192)(144 181)(157 237)(158 238)(159 239)(160 240)(161 229)(162 230)(163 231)(164 232)(165 233)(166 234)(167 235)(168 236)
(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)
(1 185 116 136)(2 135 117 184)(3 183 118 134)(4 133 119 182)(5 181 120 144)(6 143 109 192)(7 191 110 142)(8 141 111 190)(9 189 112 140)(10 139 113 188)(11 187 114 138)(12 137 115 186)(13 122 225 207)(14 206 226 121)(15 132 227 205)(16 216 228 131)(17 130 217 215)(18 214 218 129)(19 128 219 213)(20 212 220 127)(21 126 221 211)(22 210 222 125)(23 124 223 209)(24 208 224 123)(25 100 148 90)(26 89 149 99)(27 98 150 88)(28 87 151 97)(29 108 152 86)(30 85 153 107)(31 106 154 96)(32 95 155 105)(33 104 156 94)(34 93 145 103)(35 102 146 92)(36 91 147 101)(37 239 195 159)(38 158 196 238)(39 237 197 157)(40 168 198 236)(41 235 199 167)(42 166 200 234)(43 233 201 165)(44 164 202 232)(45 231 203 163)(46 162 204 230)(47 229 193 161)(48 160 194 240)(49 170 73 65)(50 64 74 169)(51 180 75 63)(52 62 76 179)(53 178 77 61)(54 72 78 177)(55 176 79 71)(56 70 80 175)(57 174 81 69)(58 68 82 173)(59 172 83 67)(60 66 84 171)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A ··· 10L 10M ··· 10T 10U 10V 10W 10X 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I ··· 20X 20Y 20Z 20AA 20AB 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 10 ··· 10 10 ··· 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 12 2 4 4 6 6 6 6 12 1 1 1 1 2 2 2 4 4 1 ··· 1 2 ··· 2 12 12 12 12 4 4 4 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D6 D6 C4○D4 D12 C5×S3 C5×D4 S3×C10 S3×C10 C5×C4○D4 C5×D12 D4⋊2S3 C5×D4⋊2S3 kernel C5×C23.21D6 C5×C4⋊Dic3 C5×D6⋊C4 C15×C22⋊C4 Dic3×C2×C10 C10×C3⋊D4 C23.21D6 C4⋊Dic3 D6⋊C4 C3×C22⋊C4 C22×Dic3 C2×C3⋊D4 C5×C22⋊C4 C2×C30 C2×C20 C22×C10 C30 C2×C10 C22⋊C4 C2×C6 C2×C4 C23 C6 C22 C10 C2 # reps 1 2 2 1 1 1 4 8 8 4 4 4 1 2 2 1 4 4 4 8 8 4 16 16 2 8

Matrix representation of C5×C23.21D6 in GL6(𝔽61)

 34 0 0 0 0 0 0 34 0 0 0 0 0 0 20 0 0 0 0 0 0 20 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 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 1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 60 0 0 0 0 0 0 0 1 60 0 0 0 0 1 0
,
 34 2 0 0 0 0 2 27 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 38 38 0 0 0 0 15 23

G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,34,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[34,2,0,0,0,0,2,27,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,38,15,0,0,0,0,38,23] >;

C5×C23.21D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{21}D_6
% in TeX

G:=Group("C5xC2^3.21D6");
// GroupNames label

G:=SmallGroup(480,765);
// by ID

G=gap.SmallGroup(480,765);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,926,891,646,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^12=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

׿
×
𝔽