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G = C4⋊Dic3⋊D5order 480 = 25·3·5

2nd semidirect product of C4⋊Dic3 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊Dic32D5, (C2×C12).6D10, (C2×C20).220D6, C6.Dic101C2, (C2×Dic5).2D6, C151(C422C2), (Dic3×Dic5)⋊2C2, D304C4.1C2, D303C4.9C2, C10.2(C4○D12), C6.63(C4○D20), C10.D415S3, C6.4(D42D5), (C2×C30).27C23, C6.2(Q82D5), (C2×Dic3).3D10, C30.102(C4○D4), C2.6(C12.28D10), C2.8(D12⋊D5), (C2×C60).313C22, C10.38(D42S3), C10.22(Q83S3), (C6×Dic5).12C22, C2.10(Dic5.D6), (C10×Dic3).13C22, (C2×Dic15).34C22, (C22×D15).16C22, C53(C4⋊C4⋊S3), C32(C4⋊C4⋊D5), (C2×C4).26(S3×D5), (C5×C4⋊Dic3)⋊14C2, C22.120(C2×S3×D5), (C2×C6).39(C22×D5), (C3×C10.D4)⋊18C2, (C2×C10).39(C22×S3), SmallGroup(480,413)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C4⋊Dic3⋊D5
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — C4⋊Dic3⋊D5
C15C2×C30 — C4⋊Dic3⋊D5
C1C22C2×C4

Generators and relations for C4⋊Dic3⋊D5
 G = < a,b,c,d,e | a4=b6=d5=e2=1, c2=b3, ab=ba, cac-1=a-1, ad=da, eae=ab3, cbc-1=ebe=b-1, bd=db, cd=dc, ece=a2c, ede=d-1 >

Subgroups: 652 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, S3, C6 [×3], C2×C4, C2×C4 [×5], C23, D5, C10 [×3], Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C15, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×3], C20 [×3], D10 [×3], C2×C10, C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15, C30 [×3], C422C2, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C3×C4⋊C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, C60, D30 [×3], C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4 [×3], C5×C4⋊C4, C4⋊C4⋊S3, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, C22×D15, C4⋊C4⋊D5, Dic3×Dic5, D304C4 [×2], C6.Dic10, C3×C10.D4, C5×C4⋊Dic3, D303C4, C4⋊Dic3⋊D5
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, C422C2, C22×D5, C4○D12, D42S3, Q83S3, S3×D5, C4○D20, D42D5, Q82D5, C4⋊C4⋊S3, C2×S3×D5, C4⋊C4⋊D5, D12⋊D5, C12.28D10, Dic5.D6, C4⋊Dic3⋊D5

Smallest permutation representation of C4⋊Dic3⋊D5
On 240 points
Generators in S240
(1 102 14 79)(2 97 15 80)(3 98 16 81)(4 99 17 82)(5 100 18 83)(6 101 13 84)(7 174 234 214)(8 169 229 215)(9 170 230 216)(10 171 231 211)(11 172 232 212)(12 173 233 213)(19 119 59 74)(20 120 60 75)(21 115 55 76)(22 116 56 77)(23 117 57 78)(24 118 58 73)(25 127 62 85)(26 128 63 86)(27 129 64 87)(28 130 65 88)(29 131 66 89)(30 132 61 90)(31 166 226 188)(32 167 227 189)(33 168 228 190)(34 163 223 191)(35 164 224 192)(36 165 225 187)(37 140 200 184)(38 141 201 185)(39 142 202 186)(40 143 203 181)(41 144 204 182)(42 139 199 183)(43 124 69 103)(44 125 70 104)(45 126 71 105)(46 121 72 106)(47 122 67 107)(48 123 68 108)(49 153 93 109)(50 154 94 110)(51 155 95 111)(52 156 96 112)(53 151 91 113)(54 152 92 114)(133 193 145 205)(134 194 146 206)(135 195 147 207)(136 196 148 208)(137 197 149 209)(138 198 150 210)(157 217 179 239)(158 218 180 240)(159 219 175 235)(160 220 176 236)(161 221 177 237)(162 222 178 238)
(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)
(1 149 4 146)(2 148 5 145)(3 147 6 150)(7 156 10 153)(8 155 11 152)(9 154 12 151)(13 138 16 135)(14 137 17 134)(15 136 18 133)(19 162 22 159)(20 161 23 158)(21 160 24 157)(25 168 28 165)(26 167 29 164)(27 166 30 163)(31 132 34 129)(32 131 35 128)(33 130 36 127)(37 126 40 123)(38 125 41 122)(39 124 42 121)(43 139 46 142)(44 144 47 141)(45 143 48 140)(49 174 52 171)(50 173 53 170)(51 172 54 169)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 191 64 188)(62 190 65 187)(63 189 66 192)(67 185 70 182)(68 184 71 181)(69 183 72 186)(73 217 76 220)(74 222 77 219)(75 221 78 218)(79 209 82 206)(80 208 83 205)(81 207 84 210)(85 228 88 225)(86 227 89 224)(87 226 90 223)(91 216 94 213)(92 215 95 212)(93 214 96 211)(97 196 100 193)(98 195 101 198)(99 194 102 197)(103 199 106 202)(104 204 107 201)(105 203 108 200)(109 234 112 231)(110 233 113 230)(111 232 114 229)(115 236 118 239)(116 235 119 238)(117 240 120 237)
(1 26 46 51 55)(2 27 47 52 56)(3 28 48 53 57)(4 29 43 54 58)(5 30 44 49 59)(6 25 45 50 60)(7 222 193 34 41)(8 217 194 35 42)(9 218 195 36 37)(10 219 196 31 38)(11 220 197 32 39)(12 221 198 33 40)(13 62 71 94 20)(14 63 72 95 21)(15 64 67 96 22)(16 65 68 91 23)(17 66 69 92 24)(18 61 70 93 19)(73 99 131 124 152)(74 100 132 125 153)(75 101 127 126 154)(76 102 128 121 155)(77 97 129 122 156)(78 98 130 123 151)(79 86 106 111 115)(80 87 107 112 116)(81 88 108 113 117)(82 89 103 114 118)(83 90 104 109 119)(84 85 105 110 120)(133 191 182 214 162)(134 192 183 215 157)(135 187 184 216 158)(136 188 185 211 159)(137 189 186 212 160)(138 190 181 213 161)(139 169 179 146 164)(140 170 180 147 165)(141 171 175 148 166)(142 172 176 149 167)(143 173 177 150 168)(144 174 178 145 163)(199 229 239 206 224)(200 230 240 207 225)(201 231 235 208 226)(202 232 236 209 227)(203 233 237 210 228)(204 234 238 205 223)
(1 55)(2 60)(3 59)(4 58)(5 57)(6 56)(7 228)(8 227)(9 226)(10 225)(11 224)(12 223)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)(25 52)(26 51)(27 50)(28 49)(29 54)(30 53)(31 230)(32 229)(33 234)(34 233)(35 232)(36 231)(37 201)(38 200)(39 199)(40 204)(41 203)(42 202)(44 48)(45 47)(61 91)(62 96)(63 95)(64 94)(65 93)(66 92)(67 71)(68 70)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 118)(80 117)(81 116)(82 115)(83 120)(84 119)(85 109)(86 114)(87 113)(88 112)(89 111)(90 110)(103 106)(104 105)(107 108)(121 124)(122 123)(125 126)(127 153)(128 152)(129 151)(130 156)(131 155)(132 154)(133 180)(134 179)(135 178)(136 177)(137 176)(138 175)(139 183)(140 182)(141 181)(142 186)(143 185)(144 184)(145 158)(146 157)(147 162)(148 161)(149 160)(150 159)(163 216)(164 215)(165 214)(166 213)(167 212)(168 211)(169 192)(170 191)(171 190)(172 189)(173 188)(174 187)(193 237)(194 236)(195 235)(196 240)(197 239)(198 238)(205 221)(206 220)(207 219)(208 218)(209 217)(210 222)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222234444444445566610···1012121212121215152020202020···2030···3060···60
size1111602466101012203030222222···2442020202044444412···124···44···4

60 irreducible representations

dim1111111222222222444444444
type+++++++++++++-++-+++
imageC1C2C2C2C2C2C2S3D5D6D6C4○D4D10D10C4○D12C4○D20D42S3Q83S3S3×D5D42D5Q82D5C2×S3×D5D12⋊D5C12.28D10Dic5.D6
kernelC4⋊Dic3⋊D5Dic3×Dic5D304C4C6.Dic10C3×C10.D4C5×C4⋊Dic3D303C4C10.D4C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12C10C6C10C10C2×C4C6C6C22C2C2C2
# reps1121111122164248112222444

Matrix representation of C4⋊Dic3⋊D5 in GL6(𝔽61)

47160000
45140000
0050000
00561100
000010
000001
,
6000000
0600000
001000
000100
0000060
0000160
,
5000000
0500000
00323000
00332900
000001
000010
,
010000
60170000
001000
000100
000010
000001
,
0600000
6000000
001000
0066000
000001
000010

G:=sub<GL(6,GF(61))| [47,45,0,0,0,0,16,14,0,0,0,0,0,0,50,56,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,32,33,0,0,0,0,30,29,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,60,0,0,0,0,1,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,6,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C4⋊Dic3⋊D5 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_3\rtimes D_5
% in TeX

G:=Group("C4:Dic3:D5");
// GroupNames label

G:=SmallGroup(480,413);
// by ID

G=gap.SmallGroup(480,413);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^6=d^5=e^2=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e=a*b^3,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=a^2*c,e*d*e=d^-1>;
// generators/relations

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