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G = C6×C23.D5order 480 = 25·3·5

Direct product of C6 and C23.D5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6×C23.D5, C10.62(C6×D4), (C23×C6).1D5, C24.2(C3×D5), (C22×C30)⋊14C4, (C2×C30).169D4, C30.419(C2×D4), (C23×C30).4C2, (C23×C10).4C6, C233(C3×Dic5), (C22×C6)⋊3Dic5, C223(C6×Dic5), C23.37(C6×D5), C3011(C22⋊C4), (C22×C10)⋊10C12, C10.41(C22×C12), C30.226(C22×C4), (C2×C30).377C23, (C22×Dic5)⋊10C6, (C6×Dic5)⋊34C22, (C22×C6).134D10, C6.38(C22×Dic5), (C22×C30).162C22, C54(C6×C22⋊C4), C2.4(C6×C5⋊D4), (C2×C30)⋊42(C2×C4), C2.9(C2×C6×Dic5), C103(C3×C22⋊C4), C1521(C2×C22⋊C4), (C2×C6×Dic5)⋊18C2, (C2×C6)⋊7(C2×Dic5), (C2×C10)⋊15(C2×C12), C22.27(D5×C2×C6), (C2×Dic5)⋊7(C2×C6), (C2×C10).44(C3×D4), C6.147(C2×C5⋊D4), (C2×C6).97(C5⋊D4), C22.25(C3×C5⋊D4), (C2×C10).60(C22×C6), (C22×C10).49(C2×C6), (C2×C6).373(C22×D5), SmallGroup(480,745)

Series: Derived Chief Lower central Upper central

C1C10 — C6×C23.D5
C1C5C10C2×C10C2×C30C6×Dic5C2×C6×Dic5 — C6×C23.D5
C5C10 — C6×C23.D5
C1C22×C6C23×C6

Generators and relations for C6×C23.D5
 G = < a,b,c,d,e,f | a6=b2=c2=d2=e5=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 592 in 264 conjugacy classes, 130 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C22, C22 [×10], C22 [×12], C5, C6, C6 [×6], C6 [×4], C2×C4 [×8], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C12 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C12 [×8], C22×C6, C22×C6 [×6], C22×C6 [×4], C30, C30 [×6], C30 [×4], C2×C22⋊C4, C2×Dic5 [×4], C2×Dic5 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C3×C22⋊C4 [×4], C22×C12 [×2], C23×C6, C3×Dic5 [×4], C2×C30, C2×C30 [×10], C2×C30 [×12], C23.D5 [×4], C22×Dic5 [×2], C23×C10, C6×C22⋊C4, C6×Dic5 [×4], C6×Dic5 [×4], C22×C30, C22×C30 [×6], C22×C30 [×4], C2×C23.D5, C3×C23.D5 [×4], C2×C6×Dic5 [×2], C23×C30, C6×C23.D5
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×4], C23, D5, C12 [×4], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C12 [×6], C3×D4 [×4], C22×C6, C3×D5, C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C3×C22⋊C4 [×4], C22×C12, C6×D4 [×2], C3×Dic5 [×4], C6×D5 [×3], C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C6×C22⋊C4, C6×Dic5 [×6], C3×C5⋊D4 [×4], D5×C2×C6, C2×C23.D5, C3×C23.D5 [×4], C2×C6×Dic5, C6×C5⋊D4 [×2], C6×C23.D5

Smallest permutation representation of C6×C23.D5
On 240 points
Generators in S240
(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)
(19 205)(20 206)(21 207)(22 208)(23 209)(24 210)(37 134)(38 135)(39 136)(40 137)(41 138)(42 133)(73 226)(74 227)(75 228)(76 223)(77 224)(78 225)(91 196)(92 197)(93 198)(94 193)(95 194)(96 195)(115 188)(116 189)(117 190)(118 191)(119 192)(120 187)(121 229)(122 230)(123 231)(124 232)(125 233)(126 234)(127 168)(128 163)(129 164)(130 165)(131 166)(132 167)(145 235)(146 236)(147 237)(148 238)(149 239)(150 240)(169 219)(170 220)(171 221)(172 222)(173 217)(174 218)(199 212)(200 213)(201 214)(202 215)(203 216)(204 211)
(1 111)(2 112)(3 113)(4 114)(5 109)(6 110)(7 70)(8 71)(9 72)(10 67)(11 68)(12 69)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(19 220)(20 221)(21 222)(22 217)(23 218)(24 219)(25 161)(26 162)(27 157)(28 158)(29 159)(30 160)(37 163)(38 164)(39 165)(40 166)(41 167)(42 168)(43 153)(44 154)(45 155)(46 156)(47 151)(48 152)(49 144)(50 139)(51 140)(52 141)(53 142)(54 143)(55 104)(56 105)(57 106)(58 107)(59 108)(60 103)(61 86)(62 87)(63 88)(64 89)(65 90)(66 85)(73 126)(74 121)(75 122)(76 123)(77 124)(78 125)(79 185)(80 186)(81 181)(82 182)(83 183)(84 184)(91 239)(92 240)(93 235)(94 236)(95 237)(96 238)(97 176)(98 177)(99 178)(100 179)(101 180)(102 175)(115 203)(116 204)(117 199)(118 200)(119 201)(120 202)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)(145 198)(146 193)(147 194)(148 195)(149 196)(150 197)(169 210)(170 205)(171 206)(172 207)(173 208)(174 209)(187 215)(188 216)(189 211)(190 212)(191 213)(192 214)(223 231)(224 232)(225 233)(226 234)(227 229)(228 230)
(1 65)(2 66)(3 61)(4 62)(5 63)(6 64)(7 44)(8 45)(9 46)(10 47)(11 48)(12 43)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 205)(20 206)(21 207)(22 208)(23 209)(24 210)(31 158)(32 159)(33 160)(34 161)(35 162)(36 157)(37 134)(38 135)(39 136)(40 137)(41 138)(42 133)(49 101)(50 102)(51 97)(52 98)(53 99)(54 100)(55 181)(56 182)(57 183)(58 184)(59 185)(60 186)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 226)(74 227)(75 228)(76 223)(77 224)(78 225)(79 108)(80 103)(81 104)(82 105)(83 106)(84 107)(85 112)(86 113)(87 114)(88 109)(89 110)(90 111)(91 196)(92 197)(93 198)(94 193)(95 194)(96 195)(115 188)(116 189)(117 190)(118 191)(119 192)(120 187)(121 229)(122 230)(123 231)(124 232)(125 233)(126 234)(127 168)(128 163)(129 164)(130 165)(131 166)(132 167)(139 175)(140 176)(141 177)(142 178)(143 179)(144 180)(145 235)(146 236)(147 237)(148 238)(149 239)(150 240)(169 219)(170 220)(171 221)(172 222)(173 217)(174 218)(199 212)(200 213)(201 214)(202 215)(203 216)(204 211)
(1 17 83 142 154)(2 18 84 143 155)(3 13 79 144 156)(4 14 80 139 151)(5 15 81 140 152)(6 16 82 141 153)(7 90 158 57 99)(8 85 159 58 100)(9 86 160 59 101)(10 87 161 60 102)(11 88 162 55 97)(12 89 157 56 98)(19 213 146 121 136)(20 214 147 122 137)(21 215 148 123 138)(22 216 149 124 133)(23 211 150 125 134)(24 212 145 126 135)(25 103 175 67 62)(26 104 176 68 63)(27 105 177 69 64)(28 106 178 70 65)(29 107 179 71 66)(30 108 180 72 61)(31 183 53 44 111)(32 184 54 45 112)(33 185 49 46 113)(34 186 50 47 114)(35 181 51 48 109)(36 182 52 43 110)(37 209 204 240 233)(38 210 199 235 234)(39 205 200 236 229)(40 206 201 237 230)(41 207 202 238 231)(42 208 203 239 232)(73 129 219 190 198)(74 130 220 191 193)(75 131 221 192 194)(76 132 222 187 195)(77 127 217 188 196)(78 128 218 189 197)(91 224 168 173 115)(92 225 163 174 116)(93 226 164 169 117)(94 227 165 170 118)(95 228 166 171 119)(96 223 167 172 120)
(1 208 111 173)(2 209 112 174)(3 210 113 169)(4 205 114 170)(5 206 109 171)(6 207 110 172)(7 188 70 216)(8 189 71 211)(9 190 72 212)(10 191 67 213)(11 192 68 214)(12 187 69 215)(13 38 33 164)(14 39 34 165)(15 40 35 166)(16 41 36 167)(17 42 31 168)(18 37 32 163)(19 87 220 62)(20 88 221 63)(21 89 222 64)(22 90 217 65)(23 85 218 66)(24 86 219 61)(25 136 161 130)(26 137 162 131)(27 138 157 132)(28 133 158 127)(29 134 159 128)(30 135 160 129)(43 120 153 202)(44 115 154 203)(45 116 155 204)(46 117 156 199)(47 118 151 200)(48 119 152 201)(49 93 144 235)(50 94 139 236)(51 95 140 237)(52 96 141 238)(53 91 142 239)(54 92 143 240)(55 75 104 122)(56 76 105 123)(57 77 106 124)(58 78 107 125)(59 73 108 126)(60 74 103 121)(79 234 185 226)(80 229 186 227)(81 230 181 228)(82 231 182 223)(83 232 183 224)(84 233 184 225)(97 194 176 147)(98 195 177 148)(99 196 178 149)(100 197 179 150)(101 198 180 145)(102 193 175 146)

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

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

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

156 conjugacy classes

class 1 2A···2G2H2I2J2K3A3B4A···4H5A5B6A···6N6O···6V10A···10AD12A···12P15A15B15C15D30A···30BH
order12···22222334···4556···66···610···1012···121515151530···30
size11···122221110···10221···12···22···210···1022222···2

156 irreducible representations

dim11111111112222222222
type++++++-+
imageC1C2C2C2C3C4C6C6C6C12D4D5Dic5D10C3×D4C3×D5C5⋊D4C3×Dic5C6×D5C3×C5⋊D4
kernelC6×C23.D5C3×C23.D5C2×C6×Dic5C23×C30C2×C23.D5C22×C30C23.D5C22×Dic5C23×C10C22×C10C2×C30C23×C6C22×C6C22×C6C2×C10C24C2×C6C23C23C22
# reps1421288421642868416161232

Matrix representation of C6×C23.D5 in GL4(𝔽61) generated by

47000
06000
00600
00060
,
60000
06000
0010
002760
,
60000
0100
0010
0001
,
1000
0100
00600
00060
,
1000
0100
00200
003658
,
11000
0100
004810
003213
G:=sub<GL(4,GF(61))| [47,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,27,0,0,0,60],[60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,20,36,0,0,0,58],[11,0,0,0,0,1,0,0,0,0,48,32,0,0,10,13] >;

C6×C23.D5 in GAP, Magma, Sage, TeX

C_6\times C_2^3.D_5
% in TeX

G:=Group("C6xC2^3.D5");
// GroupNames label

G:=SmallGroup(480,745);
// by ID

G=gap.SmallGroup(480,745);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,18822]);
// Polycyclic

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

׿
×
𝔽