metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic3⋊2D5, (C2×C12).6D10, (C2×C20).220D6, C6.Dic10⋊1C2, (C2×Dic5).2D6, C15⋊1(C42⋊2C2), (Dic3×Dic5)⋊2C2, D30⋊4C4.1C2, D30⋊3C4.9C2, C10.2(C4○D12), C6.63(C4○D20), C10.D4⋊15S3, C6.4(D4⋊2D5), (C2×C30).27C23, C6.2(Q8⋊2D5), (C2×Dic3).3D10, C30.102(C4○D4), C2.6(C12.28D10), C2.8(D12⋊D5), (C2×C60).313C22, C10.38(D4⋊2S3), C10.22(Q8⋊3S3), (C6×Dic5).12C22, C2.10(Dic5.D6), (C10×Dic3).13C22, (C2×Dic15).34C22, (C22×D15).16C22, C5⋊3(C4⋊C4⋊S3), C3⋊2(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
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, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, D30, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C4⋊C4⋊S3, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C22×D15, C4⋊C4⋊D5, Dic3×Dic5, D30⋊4C4, C6.Dic10, C3×C10.D4, C5×C4⋊Dic3, D30⋊3C4, C4⋊Dic3⋊D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C42⋊2C2, C22×D5, C4○D12, D4⋊2S3, Q8⋊3S3, S3×D5, C4○D20, D4⋊2D5, Q8⋊2D5, C4⋊C4⋊S3, C2×S3×D5, C4⋊C4⋊D5, D12⋊D5, C12.28D10, Dic5.D6, C4⋊Dic3⋊D5
►On 240 points
Generators in S240
(1 113 19 79)(2 114 20 80)(3 109 21 81)(4 110 22 82)(5 111 23 83)(6 112 24 84)(7 136 196 214)(8 137 197 215)(9 138 198 216)(10 133 193 211)(11 134 194 212)(12 135 195 213)(13 124 94 98)(14 125 95 99)(15 126 96 100)(16 121 91 101)(17 122 92 102)(18 123 93 97)(25 117 57 85)(26 118 58 86)(27 119 59 87)(28 120 60 88)(29 115 55 89)(30 116 56 90)(31 165 225 188)(32 166 226 189)(33 167 227 190)(34 168 228 191)(35 163 223 192)(36 164 224 187)(37 158 218 184)(38 159 219 185)(39 160 220 186)(40 161 221 181)(41 162 222 182)(42 157 217 183)(43 130 65 103)(44 131 66 104)(45 132 61 105)(46 127 62 106)(47 128 63 107)(48 129 64 108)(49 155 69 73)(50 156 70 74)(51 151 71 75)(52 152 72 76)(53 153 67 77)(54 154 68 78)(139 199 148 208)(140 200 149 209)(141 201 150 210)(142 202 145 205)(143 203 146 206)(144 204 147 207)(169 229 177 237)(170 230 178 238)(171 231 179 239)(172 232 180 240)(173 233 175 235)(174 234 176 236)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)(97 98 99 100 101 102)(103 104 105 106 107 108)(109 110 111 112 113 114)(115 116 117 118 119 120)(121 122 123 124 125 126)(127 128 129 130 131 132)(133 134 135 136 137 138)(139 140 141 142 143 144)(145 146 147 148 149 150)(151 152 153 154 155 156)(157 158 159 160 161 162)(163 164 165 166 167 168)(169 170 171 172 173 174)(175 176 177 178 179 180)(181 182 183 184 185 186)(187 188 189 190 191 192)(193 194 195 196 197 198)(199 200 201 202 203 204)(205 206 207 208 209 210)(211 212 213 214 215 216)(217 218 219 220 221 222)(223 224 225 226 227 228)(229 230 231 232 233 234)(235 236 237 238 239 240)
(1 149 4 146)(2 148 5 145)(3 147 6 150)(7 156 10 153)(8 155 11 152)(9 154 12 151)(13 157 16 160)(14 162 17 159)(15 161 18 158)(19 140 22 143)(20 139 23 142)(21 144 24 141)(25 172 28 169)(26 171 29 174)(27 170 30 173)(31 132 34 129)(32 131 35 128)(33 130 36 127)(37 126 40 123)(38 125 41 122)(39 124 42 121)(43 164 46 167)(44 163 47 166)(45 168 48 165)(49 134 52 137)(50 133 53 136)(51 138 54 135)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 191 64 188)(62 190 65 187)(63 189 66 192)(67 214 70 211)(68 213 71 216)(69 212 72 215)(73 194 76 197)(74 193 77 196)(75 198 78 195)(79 209 82 206)(80 208 83 205)(81 207 84 210)(85 232 88 229)(86 231 89 234)(87 230 90 233)(91 186 94 183)(92 185 95 182)(93 184 96 181)(97 218 100 221)(98 217 101 220)(99 222 102 219)(103 224 106 227)(104 223 107 226)(105 228 108 225)(109 204 112 201)(110 203 113 200)(111 202 114 199)(115 236 118 239)(116 235 119 238)(117 240 120 237)
(1 47 51 14 55)(2 48 52 15 56)(3 43 53 16 57)(4 44 54 17 58)(5 45 49 18 59)(6 46 50 13 60)(7 39 232 204 36)(8 40 233 199 31)(9 41 234 200 32)(10 42 229 201 33)(11 37 230 202 34)(12 38 231 203 35)(19 63 71 95 29)(20 64 72 96 30)(21 65 67 91 25)(22 66 68 92 26)(23 61 69 93 27)(24 62 70 94 28)(73 97 119 83 105)(74 98 120 84 106)(75 99 115 79 107)(76 100 116 80 108)(77 101 117 81 103)(78 102 118 82 104)(85 109 130 153 121)(86 110 131 154 122)(87 111 132 155 123)(88 112 127 156 124)(89 113 128 151 125)(90 114 129 152 126)(133 157 177 150 167)(134 158 178 145 168)(135 159 179 146 163)(136 160 180 147 164)(137 161 175 148 165)(138 162 176 149 166)(139 188 215 181 173)(140 189 216 182 174)(141 190 211 183 169)(142 191 212 184 170)(143 192 213 185 171)(144 187 214 186 172)(193 217 237 210 227)(194 218 238 205 228)(195 219 239 206 223)(196 220 240 207 224)(197 221 235 208 225)(198 222 236 209 226)
(1 55)(2 60)(3 59)(4 58)(5 57)(6 56)(7 197)(8 196)(9 195)(10 194)(11 193)(12 198)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 29)(20 28)(21 27)(22 26)(23 25)(24 30)(31 220)(32 219)(33 218)(34 217)(35 222)(36 221)(37 227)(38 226)(39 225)(40 224)(41 223)(42 228)(49 53)(50 52)(61 91)(62 96)(63 95)(64 94)(65 93)(66 92)(67 69)(70 72)(73 74)(75 78)(76 77)(79 118)(80 117)(81 116)(82 115)(83 120)(84 119)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(97 106)(98 105)(99 104)(100 103)(101 108)(102 107)(121 129)(122 128)(123 127)(124 132)(125 131)(126 130)(133 215)(134 214)(135 213)(136 212)(137 211)(138 216)(139 177)(140 176)(141 175)(142 180)(143 179)(144 178)(145 172)(146 171)(147 170)(148 169)(149 174)(150 173)(151 154)(152 153)(155 156)(157 188)(158 187)(159 192)(160 191)(161 190)(162 189)(163 185)(164 184)(165 183)(166 182)(167 181)(168 186)(199 240)(200 239)(201 238)(202 237)(203 236)(204 235)(205 229)(206 234)(207 233)(208 232)(209 231)(210 230)
G:=sub<Sym(240)| (1,113,19,79)(2,114,20,80)(3,109,21,81)(4,110,22,82)(5,111,23,83)(6,112,24,84)(7,136,196,214)(8,137,197,215)(9,138,198,216)(10,133,193,211)(11,134,194,212)(12,135,195,213)(13,124,94,98)(14,125,95,99)(15,126,96,100)(16,121,91,101)(17,122,92,102)(18,123,93,97)(25,117,57,85)(26,118,58,86)(27,119,59,87)(28,120,60,88)(29,115,55,89)(30,116,56,90)(31,165,225,188)(32,166,226,189)(33,167,227,190)(34,168,228,191)(35,163,223,192)(36,164,224,187)(37,158,218,184)(38,159,219,185)(39,160,220,186)(40,161,221,181)(41,162,222,182)(42,157,217,183)(43,130,65,103)(44,131,66,104)(45,132,61,105)(46,127,62,106)(47,128,63,107)(48,129,64,108)(49,155,69,73)(50,156,70,74)(51,151,71,75)(52,152,72,76)(53,153,67,77)(54,154,68,78)(139,199,148,208)(140,200,149,209)(141,201,150,210)(142,202,145,205)(143,203,146,206)(144,204,147,207)(169,229,177,237)(170,230,178,238)(171,231,179,239)(172,232,180,240)(173,233,175,235)(174,234,176,236), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192)(193,194,195,196,197,198)(199,200,201,202,203,204)(205,206,207,208,209,210)(211,212,213,214,215,216)(217,218,219,220,221,222)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,149,4,146)(2,148,5,145)(3,147,6,150)(7,156,10,153)(8,155,11,152)(9,154,12,151)(13,157,16,160)(14,162,17,159)(15,161,18,158)(19,140,22,143)(20,139,23,142)(21,144,24,141)(25,172,28,169)(26,171,29,174)(27,170,30,173)(31,132,34,129)(32,131,35,128)(33,130,36,127)(37,126,40,123)(38,125,41,122)(39,124,42,121)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,134,52,137)(50,133,53,136)(51,138,54,135)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,214,70,211)(68,213,71,216)(69,212,72,215)(73,194,76,197)(74,193,77,196)(75,198,78,195)(79,209,82,206)(80,208,83,205)(81,207,84,210)(85,232,88,229)(86,231,89,234)(87,230,90,233)(91,186,94,183)(92,185,95,182)(93,184,96,181)(97,218,100,221)(98,217,101,220)(99,222,102,219)(103,224,106,227)(104,223,107,226)(105,228,108,225)(109,204,112,201)(110,203,113,200)(111,202,114,199)(115,236,118,239)(116,235,119,238)(117,240,120,237), (1,47,51,14,55)(2,48,52,15,56)(3,43,53,16,57)(4,44,54,17,58)(5,45,49,18,59)(6,46,50,13,60)(7,39,232,204,36)(8,40,233,199,31)(9,41,234,200,32)(10,42,229,201,33)(11,37,230,202,34)(12,38,231,203,35)(19,63,71,95,29)(20,64,72,96,30)(21,65,67,91,25)(22,66,68,92,26)(23,61,69,93,27)(24,62,70,94,28)(73,97,119,83,105)(74,98,120,84,106)(75,99,115,79,107)(76,100,116,80,108)(77,101,117,81,103)(78,102,118,82,104)(85,109,130,153,121)(86,110,131,154,122)(87,111,132,155,123)(88,112,127,156,124)(89,113,128,151,125)(90,114,129,152,126)(133,157,177,150,167)(134,158,178,145,168)(135,159,179,146,163)(136,160,180,147,164)(137,161,175,148,165)(138,162,176,149,166)(139,188,215,181,173)(140,189,216,182,174)(141,190,211,183,169)(142,191,212,184,170)(143,192,213,185,171)(144,187,214,186,172)(193,217,237,210,227)(194,218,238,205,228)(195,219,239,206,223)(196,220,240,207,224)(197,221,235,208,225)(198,222,236,209,226), (1,55)(2,60)(3,59)(4,58)(5,57)(6,56)(7,197)(8,196)(9,195)(10,194)(11,193)(12,198)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,29)(20,28)(21,27)(22,26)(23,25)(24,30)(31,220)(32,219)(33,218)(34,217)(35,222)(36,221)(37,227)(38,226)(39,225)(40,224)(41,223)(42,228)(49,53)(50,52)(61,91)(62,96)(63,95)(64,94)(65,93)(66,92)(67,69)(70,72)(73,74)(75,78)(76,77)(79,118)(80,117)(81,116)(82,115)(83,120)(84,119)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(97,106)(98,105)(99,104)(100,103)(101,108)(102,107)(121,129)(122,128)(123,127)(124,132)(125,131)(126,130)(133,215)(134,214)(135,213)(136,212)(137,211)(138,216)(139,177)(140,176)(141,175)(142,180)(143,179)(144,178)(145,172)(146,171)(147,170)(148,169)(149,174)(150,173)(151,154)(152,153)(155,156)(157,188)(158,187)(159,192)(160,191)(161,190)(162,189)(163,185)(164,184)(165,183)(166,182)(167,181)(168,186)(199,240)(200,239)(201,238)(202,237)(203,236)(204,235)(205,229)(206,234)(207,233)(208,232)(209,231)(210,230)>;
G:=Group( (1,113,19,79)(2,114,20,80)(3,109,21,81)(4,110,22,82)(5,111,23,83)(6,112,24,84)(7,136,196,214)(8,137,197,215)(9,138,198,216)(10,133,193,211)(11,134,194,212)(12,135,195,213)(13,124,94,98)(14,125,95,99)(15,126,96,100)(16,121,91,101)(17,122,92,102)(18,123,93,97)(25,117,57,85)(26,118,58,86)(27,119,59,87)(28,120,60,88)(29,115,55,89)(30,116,56,90)(31,165,225,188)(32,166,226,189)(33,167,227,190)(34,168,228,191)(35,163,223,192)(36,164,224,187)(37,158,218,184)(38,159,219,185)(39,160,220,186)(40,161,221,181)(41,162,222,182)(42,157,217,183)(43,130,65,103)(44,131,66,104)(45,132,61,105)(46,127,62,106)(47,128,63,107)(48,129,64,108)(49,155,69,73)(50,156,70,74)(51,151,71,75)(52,152,72,76)(53,153,67,77)(54,154,68,78)(139,199,148,208)(140,200,149,209)(141,201,150,210)(142,202,145,205)(143,203,146,206)(144,204,147,207)(169,229,177,237)(170,230,178,238)(171,231,179,239)(172,232,180,240)(173,233,175,235)(174,234,176,236), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192)(193,194,195,196,197,198)(199,200,201,202,203,204)(205,206,207,208,209,210)(211,212,213,214,215,216)(217,218,219,220,221,222)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240), (1,149,4,146)(2,148,5,145)(3,147,6,150)(7,156,10,153)(8,155,11,152)(9,154,12,151)(13,157,16,160)(14,162,17,159)(15,161,18,158)(19,140,22,143)(20,139,23,142)(21,144,24,141)(25,172,28,169)(26,171,29,174)(27,170,30,173)(31,132,34,129)(32,131,35,128)(33,130,36,127)(37,126,40,123)(38,125,41,122)(39,124,42,121)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,134,52,137)(50,133,53,136)(51,138,54,135)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,214,70,211)(68,213,71,216)(69,212,72,215)(73,194,76,197)(74,193,77,196)(75,198,78,195)(79,209,82,206)(80,208,83,205)(81,207,84,210)(85,232,88,229)(86,231,89,234)(87,230,90,233)(91,186,94,183)(92,185,95,182)(93,184,96,181)(97,218,100,221)(98,217,101,220)(99,222,102,219)(103,224,106,227)(104,223,107,226)(105,228,108,225)(109,204,112,201)(110,203,113,200)(111,202,114,199)(115,236,118,239)(116,235,119,238)(117,240,120,237), (1,47,51,14,55)(2,48,52,15,56)(3,43,53,16,57)(4,44,54,17,58)(5,45,49,18,59)(6,46,50,13,60)(7,39,232,204,36)(8,40,233,199,31)(9,41,234,200,32)(10,42,229,201,33)(11,37,230,202,34)(12,38,231,203,35)(19,63,71,95,29)(20,64,72,96,30)(21,65,67,91,25)(22,66,68,92,26)(23,61,69,93,27)(24,62,70,94,28)(73,97,119,83,105)(74,98,120,84,106)(75,99,115,79,107)(76,100,116,80,108)(77,101,117,81,103)(78,102,118,82,104)(85,109,130,153,121)(86,110,131,154,122)(87,111,132,155,123)(88,112,127,156,124)(89,113,128,151,125)(90,114,129,152,126)(133,157,177,150,167)(134,158,178,145,168)(135,159,179,146,163)(136,160,180,147,164)(137,161,175,148,165)(138,162,176,149,166)(139,188,215,181,173)(140,189,216,182,174)(141,190,211,183,169)(142,191,212,184,170)(143,192,213,185,171)(144,187,214,186,172)(193,217,237,210,227)(194,218,238,205,228)(195,219,239,206,223)(196,220,240,207,224)(197,221,235,208,225)(198,222,236,209,226), (1,55)(2,60)(3,59)(4,58)(5,57)(6,56)(7,197)(8,196)(9,195)(10,194)(11,193)(12,198)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,29)(20,28)(21,27)(22,26)(23,25)(24,30)(31,220)(32,219)(33,218)(34,217)(35,222)(36,221)(37,227)(38,226)(39,225)(40,224)(41,223)(42,228)(49,53)(50,52)(61,91)(62,96)(63,95)(64,94)(65,93)(66,92)(67,69)(70,72)(73,74)(75,78)(76,77)(79,118)(80,117)(81,116)(82,115)(83,120)(84,119)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(97,106)(98,105)(99,104)(100,103)(101,108)(102,107)(121,129)(122,128)(123,127)(124,132)(125,131)(126,130)(133,215)(134,214)(135,213)(136,212)(137,211)(138,216)(139,177)(140,176)(141,175)(142,180)(143,179)(144,178)(145,172)(146,171)(147,170)(148,169)(149,174)(150,173)(151,154)(152,153)(155,156)(157,188)(158,187)(159,192)(160,191)(161,190)(162,189)(163,185)(164,184)(165,183)(166,182)(167,181)(168,186)(199,240)(200,239)(201,238)(202,237)(203,236)(204,235)(205,229)(206,234)(207,233)(208,232)(209,231)(210,230) );
G=PermutationGroup([[(1,113,19,79),(2,114,20,80),(3,109,21,81),(4,110,22,82),(5,111,23,83),(6,112,24,84),(7,136,196,214),(8,137,197,215),(9,138,198,216),(10,133,193,211),(11,134,194,212),(12,135,195,213),(13,124,94,98),(14,125,95,99),(15,126,96,100),(16,121,91,101),(17,122,92,102),(18,123,93,97),(25,117,57,85),(26,118,58,86),(27,119,59,87),(28,120,60,88),(29,115,55,89),(30,116,56,90),(31,165,225,188),(32,166,226,189),(33,167,227,190),(34,168,228,191),(35,163,223,192),(36,164,224,187),(37,158,218,184),(38,159,219,185),(39,160,220,186),(40,161,221,181),(41,162,222,182),(42,157,217,183),(43,130,65,103),(44,131,66,104),(45,132,61,105),(46,127,62,106),(47,128,63,107),(48,129,64,108),(49,155,69,73),(50,156,70,74),(51,151,71,75),(52,152,72,76),(53,153,67,77),(54,154,68,78),(139,199,148,208),(140,200,149,209),(141,201,150,210),(142,202,145,205),(143,203,146,206),(144,204,147,207),(169,229,177,237),(170,230,178,238),(171,231,179,239),(172,232,180,240),(173,233,175,235),(174,234,176,236)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96),(97,98,99,100,101,102),(103,104,105,106,107,108),(109,110,111,112,113,114),(115,116,117,118,119,120),(121,122,123,124,125,126),(127,128,129,130,131,132),(133,134,135,136,137,138),(139,140,141,142,143,144),(145,146,147,148,149,150),(151,152,153,154,155,156),(157,158,159,160,161,162),(163,164,165,166,167,168),(169,170,171,172,173,174),(175,176,177,178,179,180),(181,182,183,184,185,186),(187,188,189,190,191,192),(193,194,195,196,197,198),(199,200,201,202,203,204),(205,206,207,208,209,210),(211,212,213,214,215,216),(217,218,219,220,221,222),(223,224,225,226,227,228),(229,230,231,232,233,234),(235,236,237,238,239,240)], [(1,149,4,146),(2,148,5,145),(3,147,6,150),(7,156,10,153),(8,155,11,152),(9,154,12,151),(13,157,16,160),(14,162,17,159),(15,161,18,158),(19,140,22,143),(20,139,23,142),(21,144,24,141),(25,172,28,169),(26,171,29,174),(27,170,30,173),(31,132,34,129),(32,131,35,128),(33,130,36,127),(37,126,40,123),(38,125,41,122),(39,124,42,121),(43,164,46,167),(44,163,47,166),(45,168,48,165),(49,134,52,137),(50,133,53,136),(51,138,54,135),(55,176,58,179),(56,175,59,178),(57,180,60,177),(61,191,64,188),(62,190,65,187),(63,189,66,192),(67,214,70,211),(68,213,71,216),(69,212,72,215),(73,194,76,197),(74,193,77,196),(75,198,78,195),(79,209,82,206),(80,208,83,205),(81,207,84,210),(85,232,88,229),(86,231,89,234),(87,230,90,233),(91,186,94,183),(92,185,95,182),(93,184,96,181),(97,218,100,221),(98,217,101,220),(99,222,102,219),(103,224,106,227),(104,223,107,226),(105,228,108,225),(109,204,112,201),(110,203,113,200),(111,202,114,199),(115,236,118,239),(116,235,119,238),(117,240,120,237)], [(1,47,51,14,55),(2,48,52,15,56),(3,43,53,16,57),(4,44,54,17,58),(5,45,49,18,59),(6,46,50,13,60),(7,39,232,204,36),(8,40,233,199,31),(9,41,234,200,32),(10,42,229,201,33),(11,37,230,202,34),(12,38,231,203,35),(19,63,71,95,29),(20,64,72,96,30),(21,65,67,91,25),(22,66,68,92,26),(23,61,69,93,27),(24,62,70,94,28),(73,97,119,83,105),(74,98,120,84,106),(75,99,115,79,107),(76,100,116,80,108),(77,101,117,81,103),(78,102,118,82,104),(85,109,130,153,121),(86,110,131,154,122),(87,111,132,155,123),(88,112,127,156,124),(89,113,128,151,125),(90,114,129,152,126),(133,157,177,150,167),(134,158,178,145,168),(135,159,179,146,163),(136,160,180,147,164),(137,161,175,148,165),(138,162,176,149,166),(139,188,215,181,173),(140,189,216,182,174),(141,190,211,183,169),(142,191,212,184,170),(143,192,213,185,171),(144,187,214,186,172),(193,217,237,210,227),(194,218,238,205,228),(195,219,239,206,223),(196,220,240,207,224),(197,221,235,208,225),(198,222,236,209,226)], [(1,55),(2,60),(3,59),(4,58),(5,57),(6,56),(7,197),(8,196),(9,195),(10,194),(11,193),(12,198),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,29),(20,28),(21,27),(22,26),(23,25),(24,30),(31,220),(32,219),(33,218),(34,217),(35,222),(36,221),(37,227),(38,226),(39,225),(40,224),(41,223),(42,228),(49,53),(50,52),(61,91),(62,96),(63,95),(64,94),(65,93),(66,92),(67,69),(70,72),(73,74),(75,78),(76,77),(79,118),(80,117),(81,116),(82,115),(83,120),(84,119),(85,114),(86,113),(87,112),(88,111),(89,110),(90,109),(97,106),(98,105),(99,104),(100,103),(101,108),(102,107),(121,129),(122,128),(123,127),(124,132),(125,131),(126,130),(133,215),(134,214),(135,213),(136,212),(137,211),(138,216),(139,177),(140,176),(141,175),(142,180),(143,179),(144,178),(145,172),(146,171),(147,170),(148,169),(149,174),(150,173),(151,154),(152,153),(155,156),(157,188),(158,187),(159,192),(160,191),(161,190),(162,189),(163,185),(164,184),(165,183),(166,182),(167,181),(168,186),(199,240),(200,239),(201,238),(202,237),(203,236),(204,235),(205,229),(206,234),(207,233),(208,232),(209,231),(210,230)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 60 | 2 | 4 | 6 | 6 | 10 | 10 | 12 | 20 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4○D12 | C4○D20 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | Q8⋊2D5 | C2×S3×D5 | D12⋊D5 | C12.28D10 | Dic5.D6 |
| kernel | C4⋊Dic3⋊D5 | Dic3×Dic5 | D30⋊4C4 | C6.Dic10 | C3×C10.D4 | C5×C4⋊Dic3 | D30⋊3C4 | C10.D4 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 6 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C4⋊Dic3⋊D5 ►in GL6(𝔽61)
| 47 | 16 | 0 | 0 | 0 | 0 |
| 45 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 56 | 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 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 33 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 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 | 0 | 0 | 0 |
| 0 | 0 | 6 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
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