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G = (C2×D12).D5order 480 = 25·3·5

4th non-split extension by C2×D12 of D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D12).4D5, (S3×C10).5D4, D6⋊Dic510C2, (C2×C20).227D6, C30.135(C2×D4), C10.131(S3×D4), (C2×C12).20D10, C57(D6.D4), D6.5(C5⋊D4), (C10×D12).11C2, C30.65(C4○D4), C10.D418S3, C30.4Q819C2, C10.57(C4○D12), C6.25(D42D5), (C2×C60).320C22, (C2×C30).113C23, (C2×Dic5).109D6, (C22×S3).40D10, C159(C22.D4), C2.17(D12⋊D5), C2.13(D125D5), C10.34(Q83S3), C31(C23.18D10), (C6×Dic5).67C22, (C2×Dic15).91C22, (C2×S3×Dic5)⋊5C2, (C2×C4).49(S3×D5), C6.32(C2×C5⋊D4), C2.13(S3×C5⋊D4), C22.179(C2×S3×D5), (S3×C2×C10).21C22, (C3×C10.D4)⋊21C2, (C2×C6).125(C22×D5), (C2×C10).125(C22×S3), SmallGroup(480,499)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C2×D12).D5
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — (C2×D12).D5
C15C2×C30 — (C2×D12).D5
C1C22C2×C4

Generators and relations for (C2×D12).D5
 G = < a,b,c,d,e | a10=b3=c2=d4=1, e2=a5, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=b-1, bd=db, be=eb, dcd-1=a5c, ce=ec, ede-1=d-1 >

Subgroups: 716 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3 [×3], C6 [×3], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C10 [×3], C10 [×3], Dic3 [×2], C12 [×3], D6 [×2], D6 [×5], C2×C6, C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×4], C20, C2×C10, C2×C10 [×7], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C5×S3 [×3], C30 [×3], C22.D4, C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], Dic3⋊C4, D6⋊C4 [×3], C3×C4⋊C4, S3×C2×C4, C2×D12, C3×Dic5 [×2], Dic15 [×2], C60, S3×C10 [×2], S3×C10 [×5], C2×C30, C10.D4, C10.D4, C23.D5 [×3], C22×Dic5, D4×C10, D6.D4, S3×Dic5 [×2], C6×Dic5 [×2], C5×D12 [×2], C2×Dic15 [×2], C2×C60, S3×C2×C10 [×2], C23.18D10, D6⋊Dic5 [×3], C3×C10.D4, C30.4Q8, C2×S3×Dic5, C10×D12, (C2×D12).D5
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C22.D4, C5⋊D4 [×2], C22×D5, C4○D12, S3×D4, Q83S3, S3×D5, D42D5 [×2], C2×C5⋊D4, D6.D4, C2×S3×D5, C23.18D10, D12⋊D5, D125D5, S3×C5⋊D4, (C2×D12).D5

Smallest permutation representation of (C2×D12).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)
(1 40 25)(2 31 26)(3 32 27)(4 33 28)(5 34 29)(6 35 30)(7 36 21)(8 37 22)(9 38 23)(10 39 24)(11 229 232)(12 230 233)(13 221 234)(14 222 235)(15 223 236)(16 224 237)(17 225 238)(18 226 239)(19 227 240)(20 228 231)(41 53 66)(42 54 67)(43 55 68)(44 56 69)(45 57 70)(46 58 61)(47 59 62)(48 60 63)(49 51 64)(50 52 65)(71 83 96)(72 84 97)(73 85 98)(74 86 99)(75 87 100)(76 88 91)(77 89 92)(78 90 93)(79 81 94)(80 82 95)(101 113 125)(102 114 126)(103 115 127)(104 116 128)(105 117 129)(106 118 130)(107 119 121)(108 120 122)(109 111 123)(110 112 124)(131 144 156)(132 145 157)(133 146 158)(134 147 159)(135 148 160)(136 149 151)(137 150 152)(138 141 153)(139 142 154)(140 143 155)(161 174 186)(162 175 187)(163 176 188)(164 177 189)(165 178 190)(166 179 181)(167 180 182)(168 171 183)(169 172 184)(170 173 185)(191 204 216)(192 205 217)(193 206 218)(194 207 219)(195 208 220)(196 209 211)(197 210 212)(198 201 213)(199 202 214)(200 203 215)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 199)(12 200)(13 191)(14 192)(15 193)(16 194)(17 195)(18 196)(19 197)(20 198)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 121)(78 122)(79 123)(80 124)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(131 181)(132 182)(133 183)(134 184)(135 185)(136 186)(137 187)(138 188)(139 189)(140 190)(141 176)(142 177)(143 178)(144 179)(145 180)(146 171)(147 172)(148 173)(149 174)(150 175)(151 161)(152 162)(153 163)(154 164)(155 165)(156 166)(157 167)(158 168)(159 169)(160 170)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)(211 226)(212 227)(213 228)(214 229)(215 230)(216 221)(217 222)(218 223)(219 224)(220 225)
(1 233 57 203)(2 232 58 202)(3 231 59 201)(4 240 60 210)(5 239 51 209)(6 238 52 208)(7 237 53 207)(8 236 54 206)(9 235 55 205)(10 234 56 204)(11 61 214 31)(12 70 215 40)(13 69 216 39)(14 68 217 38)(15 67 218 37)(16 66 219 36)(17 65 220 35)(18 64 211 34)(19 63 212 33)(20 62 213 32)(21 224 41 194)(22 223 42 193)(23 222 43 192)(24 221 44 191)(25 230 45 200)(26 229 46 199)(27 228 47 198)(28 227 48 197)(29 226 49 196)(30 225 50 195)(71 169 101 139)(72 168 102 138)(73 167 103 137)(74 166 104 136)(75 165 105 135)(76 164 106 134)(77 163 107 133)(78 162 108 132)(79 161 109 131)(80 170 110 140)(81 174 111 144)(82 173 112 143)(83 172 113 142)(84 171 114 141)(85 180 115 150)(86 179 116 149)(87 178 117 148)(88 177 118 147)(89 176 119 146)(90 175 120 145)(91 189 130 159)(92 188 121 158)(93 187 122 157)(94 186 123 156)(95 185 124 155)(96 184 125 154)(97 183 126 153)(98 182 127 152)(99 181 128 151)(100 190 129 160)
(1 148 6 143)(2 147 7 142)(3 146 8 141)(4 145 9 150)(5 144 10 149)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 139 26 134)(22 138 27 133)(23 137 28 132)(24 136 29 131)(25 135 30 140)(31 159 36 154)(32 158 37 153)(33 157 38 152)(34 156 39 151)(35 155 40 160)(41 169 46 164)(42 168 47 163)(43 167 48 162)(44 166 49 161)(45 165 50 170)(51 174 56 179)(52 173 57 178)(53 172 58 177)(54 171 59 176)(55 180 60 175)(61 189 66 184)(62 188 67 183)(63 187 68 182)(64 186 69 181)(65 185 70 190)(71 199 76 194)(72 198 77 193)(73 197 78 192)(74 196 79 191)(75 195 80 200)(81 204 86 209)(82 203 87 208)(83 202 88 207)(84 201 89 206)(85 210 90 205)(91 219 96 214)(92 218 97 213)(93 217 98 212)(94 216 99 211)(95 215 100 220)(101 229 106 224)(102 228 107 223)(103 227 108 222)(104 226 109 221)(105 225 110 230)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E12F15A15B20A20B20C20D30A···30F60A···60H
order1222222344444445566610···1010···1012121212121215152020202030···3060···60
size1111661224101020303060222222···212···1244202020204444444···44···4

60 irreducible representations

dim111111222222222244444444
type++++++++++++++++-+-
imageC1C2C2C2C2C2S3D4D5D6D6C4○D4D10D10C5⋊D4C4○D12S3×D4Q83S3S3×D5D42D5C2×S3×D5D12⋊D5D125D5S3×C5⋊D4
kernel(C2×D12).D5D6⋊Dic5C3×C10.D4C30.4Q8C2×S3×Dic5C10×D12C10.D4S3×C10C2×D12C2×Dic5C2×C20C30C2×C12C22×S3D6C10C10C10C2×C4C6C22C2C2C2
# reps131111122214248411242444

Matrix representation of (C2×D12).D5 in GL4(𝔽61) generated by

60000
06000
001743
00170
,
0100
606000
0010
0001
,
60000
1100
00600
00060
,
524300
18900
00533
00198
,
11000
01100
002856
005933
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,17,17,0,0,43,0],[0,60,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[52,18,0,0,43,9,0,0,0,0,53,19,0,0,3,8],[11,0,0,0,0,11,0,0,0,0,28,59,0,0,56,33] >;

(C2×D12).D5 in GAP, Magma, Sage, TeX

(C_2\times D_{12}).D_5
% in TeX

G:=Group("(C2xD12).D5");
// GroupNames label

G:=SmallGroup(480,499);
// by ID

G=gap.SmallGroup(480,499);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^3=c^2=d^4=1,e^2=a^5,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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