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## G = C5×C23.12D6order 480 = 25·3·5

### Direct product of C5 and C23.12D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C23.12D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C20 — C5×C23.12D6
 Lower central C3 — C2×C6 — C5×C23.12D6
 Upper central C1 — C2×C10 — D4×C10

Generators and relations for C5×C23.12D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 324 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], Dic6 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×2], C30 [×2], C4.4D4, C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×C10 [×2], C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, C5×Dic3 [×4], C60 [×2], C2×C30, C2×C30 [×6], C4×C20, C5×C22⋊C4 [×4], D4×C10, Q8×C10, C23.12D6, C5×Dic6 [×2], C10×Dic3 [×4], C2×C60, D4×C15 [×2], C22×C30 [×2], C5×C4.4D4, Dic3×C20, C5×C6.D4 [×4], C10×Dic6, D4×C30, C5×C23.12D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C4○D4 [×2], C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C4.4D4, C5×D4 [×2], C22×C10, D42S3 [×2], C2×C3⋊D4, S3×C10 [×3], D4×C10, C5×C4○D4 [×2], C23.12D6, C5×C3⋊D4 [×2], S3×C2×C10, C5×C4.4D4, C5×D42S3 [×2], C10×C3⋊D4, C5×C23.12D6

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

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

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

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

120 conjugacy classes

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

120 irreducible representations

 dim 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 C5 C10 C10 C10 C10 S3 D4 D6 D6 C4○D4 C3⋊D4 C5×S3 C5×D4 S3×C10 S3×C10 C5×C4○D4 C5×C3⋊D4 D4⋊2S3 C5×D4⋊2S3 kernel C5×C23.12D6 Dic3×C20 C5×C6.D4 C10×Dic6 D4×C30 C23.12D6 C4×Dic3 C6.D4 C2×Dic6 C6×D4 D4×C10 C60 C2×C20 C22×C10 C30 C20 C2×D4 C12 C2×C4 C23 C6 C4 C10 C2 # reps 1 1 4 1 1 4 4 16 4 4 1 2 1 2 4 4 4 8 4 8 16 16 2 8

Matrix representation of C5×C23.12D6 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 10 60 0 0 0 0 60 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 13 0 0 0 13 47 0 0 0 0 0 60 0 0 1 0
,
 47 15 0 0 56 14 0 0 0 0 50 0 0 0 0 50
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,10,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[13,13,0,0,0,47,0,0,0,0,0,1,0,0,60,0],[47,56,0,0,15,14,0,0,0,0,50,0,0,0,0,50] >;

C5×C23.12D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{12}D_6
% in TeX

G:=Group("C5xC2^3.12D6");
// GroupNames label

G:=SmallGroup(480,815);
// by ID

G=gap.SmallGroup(480,815);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,288,2606,471,15686]);
// Polycyclic

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

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