metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic5⋊3S3, (C2×C20).9D6, Dic3⋊C4⋊15D5, C6.5(C4○D20), C30.Q8⋊2C2, (C2×Dic5).6D6, C15⋊4(C42⋊2C2), (Dic3×Dic5)⋊6C2, D30⋊3C4.6C2, D30⋊4C4.2C2, C30.14(C4○D4), (C2×C12).219D10, (C2×C30).35C23, (C2×Dic3).4D10, C10.67(C4○D12), C2.9(D60⋊C2), C2.6(D20⋊S3), C6.36(D4⋊2D5), C10.2(D4⋊2S3), (C2×C60).250C22, C6.21(Q8⋊2D5), C10.6(Q8⋊3S3), (C6×Dic5).20C22, C2.10(Dic3.D10), (C2×Dic15).41C22, (C10×Dic3).19C22, (C22×D15).17C22, C5⋊2(C4⋊C4⋊S3), C3⋊3(C4⋊C4⋊D5), (C2×C4).31(S3×D5), (C3×C4⋊Dic5)⋊15C2, C22.125(C2×S3×D5), (C5×Dic3⋊C4)⋊15C2, (C2×C6).47(C22×D5), (C2×C10).47(C22×S3), SmallGroup(480,421)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic5⋊S3
G = < a,b,c,d,e | a4=b10=d3=e2=1, c2=b5, ab=ba, cac-1=a-1, ad=da, eae=ab5, 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, C30.Q8, C3×C4⋊Dic5, C5×Dic3⋊C4, D30⋊3C4, C4⋊Dic5⋊S3
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, D20⋊S3, D60⋊C2, Dic3.D10, C4⋊Dic5⋊S3
►On 240 points
Generators in S240
(1 116 56 77)(2 117 57 78)(3 118 58 79)(4 119 59 80)(5 120 60 71)(6 111 51 72)(7 112 52 73)(8 113 53 74)(9 114 54 75)(10 115 55 76)(11 90 30 94)(12 81 21 95)(13 82 22 96)(14 83 23 97)(15 84 24 98)(16 85 25 99)(17 86 26 100)(18 87 27 91)(19 88 28 92)(20 89 29 93)(31 169 224 190)(32 170 225 181)(33 161 226 182)(34 162 227 183)(35 163 228 184)(36 164 229 185)(37 165 230 186)(38 166 221 187)(39 167 222 188)(40 168 223 189)(41 122 62 101)(42 123 63 102)(43 124 64 103)(44 125 65 104)(45 126 66 105)(46 127 67 106)(47 128 68 107)(48 129 69 108)(49 130 70 109)(50 121 61 110)(131 196 180 235)(132 197 171 236)(133 198 172 237)(134 199 173 238)(135 200 174 239)(136 191 175 240)(137 192 176 231)(138 193 177 232)(139 194 178 233)(140 195 179 234)(141 206 155 220)(142 207 156 211)(143 208 157 212)(144 209 158 213)(145 210 159 214)(146 201 160 215)(147 202 151 216)(148 203 152 217)(149 204 153 218)(150 205 154 219)
(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)
(1 197 6 192)(2 196 7 191)(3 195 8 200)(4 194 9 199)(5 193 10 198)(11 35 16 40)(12 34 17 39)(13 33 18 38)(14 32 19 37)(15 31 20 36)(21 227 26 222)(22 226 27 221)(23 225 28 230)(24 224 29 229)(25 223 30 228)(41 207 46 202)(42 206 47 201)(43 205 48 210)(44 204 49 209)(45 203 50 208)(51 231 56 236)(52 240 57 235)(53 239 58 234)(54 238 59 233)(55 237 60 232)(61 212 66 217)(62 211 67 216)(63 220 68 215)(64 219 69 214)(65 218 70 213)(71 177 76 172)(72 176 77 171)(73 175 78 180)(74 174 79 179)(75 173 80 178)(81 183 86 188)(82 182 87 187)(83 181 88 186)(84 190 89 185)(85 189 90 184)(91 166 96 161)(92 165 97 170)(93 164 98 169)(94 163 99 168)(95 162 100 167)(101 156 106 151)(102 155 107 160)(103 154 108 159)(104 153 109 158)(105 152 110 157)(111 137 116 132)(112 136 117 131)(113 135 118 140)(114 134 119 139)(115 133 120 138)(121 143 126 148)(122 142 127 147)(123 141 128 146)(124 150 129 145)(125 149 130 144)
(1 27 41)(2 28 42)(3 29 43)(4 30 44)(5 21 45)(6 22 46)(7 23 47)(8 24 48)(9 25 49)(10 26 50)(11 65 59)(12 66 60)(13 67 51)(14 68 52)(15 69 53)(16 70 54)(17 61 55)(18 62 56)(19 63 57)(20 64 58)(31 214 239)(32 215 240)(33 216 231)(34 217 232)(35 218 233)(36 219 234)(37 220 235)(38 211 236)(39 212 237)(40 213 238)(71 81 105)(72 82 106)(73 83 107)(74 84 108)(75 85 109)(76 86 110)(77 87 101)(78 88 102)(79 89 103)(80 90 104)(91 122 116)(92 123 117)(93 124 118)(94 125 119)(95 126 120)(96 127 111)(97 128 112)(98 129 113)(99 130 114)(100 121 115)(131 165 141)(132 166 142)(133 167 143)(134 168 144)(135 169 145)(136 170 146)(137 161 147)(138 162 148)(139 163 149)(140 164 150)(151 176 182)(152 177 183)(153 178 184)(154 179 185)(155 180 186)(156 171 187)(157 172 188)(158 173 189)(159 174 190)(160 175 181)(191 225 201)(192 226 202)(193 227 203)(194 228 204)(195 229 205)(196 230 206)(197 221 207)(198 222 208)(199 223 209)(200 224 210)
(2 10)(3 9)(4 8)(5 7)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 70)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 50)(29 49)(30 48)(31 204)(32 203)(33 202)(34 201)(35 210)(36 209)(37 208)(38 207)(39 206)(40 205)(52 60)(53 59)(54 58)(55 57)(71 78)(72 77)(73 76)(74 75)(79 80)(81 102)(82 101)(83 110)(84 109)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 127)(92 126)(93 125)(94 124)(95 123)(96 122)(97 121)(98 130)(99 129)(100 128)(111 116)(112 115)(113 114)(117 120)(118 119)(131 177)(132 176)(133 175)(134 174)(135 173)(136 172)(137 171)(138 180)(139 179)(140 178)(141 183)(142 182)(143 181)(144 190)(145 189)(146 188)(147 187)(148 186)(149 185)(150 184)(151 166)(152 165)(153 164)(154 163)(155 162)(156 161)(157 170)(158 169)(159 168)(160 167)(191 232)(192 231)(193 240)(194 239)(195 238)(196 237)(197 236)(198 235)(199 234)(200 233)(211 221)(212 230)(213 229)(214 228)(215 227)(216 226)(217 225)(218 224)(219 223)(220 222)
G:=sub<Sym(240)| (1,116,56,77)(2,117,57,78)(3,118,58,79)(4,119,59,80)(5,120,60,71)(6,111,51,72)(7,112,52,73)(8,113,53,74)(9,114,54,75)(10,115,55,76)(11,90,30,94)(12,81,21,95)(13,82,22,96)(14,83,23,97)(15,84,24,98)(16,85,25,99)(17,86,26,100)(18,87,27,91)(19,88,28,92)(20,89,29,93)(31,169,224,190)(32,170,225,181)(33,161,226,182)(34,162,227,183)(35,163,228,184)(36,164,229,185)(37,165,230,186)(38,166,221,187)(39,167,222,188)(40,168,223,189)(41,122,62,101)(42,123,63,102)(43,124,64,103)(44,125,65,104)(45,126,66,105)(46,127,67,106)(47,128,68,107)(48,129,69,108)(49,130,70,109)(50,121,61,110)(131,196,180,235)(132,197,171,236)(133,198,172,237)(134,199,173,238)(135,200,174,239)(136,191,175,240)(137,192,176,231)(138,193,177,232)(139,194,178,233)(140,195,179,234)(141,206,155,220)(142,207,156,211)(143,208,157,212)(144,209,158,213)(145,210,159,214)(146,201,160,215)(147,202,151,216)(148,203,152,217)(149,204,153,218)(150,205,154,219), (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), (1,197,6,192)(2,196,7,191)(3,195,8,200)(4,194,9,199)(5,193,10,198)(11,35,16,40)(12,34,17,39)(13,33,18,38)(14,32,19,37)(15,31,20,36)(21,227,26,222)(22,226,27,221)(23,225,28,230)(24,224,29,229)(25,223,30,228)(41,207,46,202)(42,206,47,201)(43,205,48,210)(44,204,49,209)(45,203,50,208)(51,231,56,236)(52,240,57,235)(53,239,58,234)(54,238,59,233)(55,237,60,232)(61,212,66,217)(62,211,67,216)(63,220,68,215)(64,219,69,214)(65,218,70,213)(71,177,76,172)(72,176,77,171)(73,175,78,180)(74,174,79,179)(75,173,80,178)(81,183,86,188)(82,182,87,187)(83,181,88,186)(84,190,89,185)(85,189,90,184)(91,166,96,161)(92,165,97,170)(93,164,98,169)(94,163,99,168)(95,162,100,167)(101,156,106,151)(102,155,107,160)(103,154,108,159)(104,153,109,158)(105,152,110,157)(111,137,116,132)(112,136,117,131)(113,135,118,140)(114,134,119,139)(115,133,120,138)(121,143,126,148)(122,142,127,147)(123,141,128,146)(124,150,129,145)(125,149,130,144), (1,27,41)(2,28,42)(3,29,43)(4,30,44)(5,21,45)(6,22,46)(7,23,47)(8,24,48)(9,25,49)(10,26,50)(11,65,59)(12,66,60)(13,67,51)(14,68,52)(15,69,53)(16,70,54)(17,61,55)(18,62,56)(19,63,57)(20,64,58)(31,214,239)(32,215,240)(33,216,231)(34,217,232)(35,218,233)(36,219,234)(37,220,235)(38,211,236)(39,212,237)(40,213,238)(71,81,105)(72,82,106)(73,83,107)(74,84,108)(75,85,109)(76,86,110)(77,87,101)(78,88,102)(79,89,103)(80,90,104)(91,122,116)(92,123,117)(93,124,118)(94,125,119)(95,126,120)(96,127,111)(97,128,112)(98,129,113)(99,130,114)(100,121,115)(131,165,141)(132,166,142)(133,167,143)(134,168,144)(135,169,145)(136,170,146)(137,161,147)(138,162,148)(139,163,149)(140,164,150)(151,176,182)(152,177,183)(153,178,184)(154,179,185)(155,180,186)(156,171,187)(157,172,188)(158,173,189)(159,174,190)(160,175,181)(191,225,201)(192,226,202)(193,227,203)(194,228,204)(195,229,205)(196,230,206)(197,221,207)(198,222,208)(199,223,209)(200,224,210), (2,10)(3,9)(4,8)(5,7)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,70)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,50)(29,49)(30,48)(31,204)(32,203)(33,202)(34,201)(35,210)(36,209)(37,208)(38,207)(39,206)(40,205)(52,60)(53,59)(54,58)(55,57)(71,78)(72,77)(73,76)(74,75)(79,80)(81,102)(82,101)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,130)(99,129)(100,128)(111,116)(112,115)(113,114)(117,120)(118,119)(131,177)(132,176)(133,175)(134,174)(135,173)(136,172)(137,171)(138,180)(139,179)(140,178)(141,183)(142,182)(143,181)(144,190)(145,189)(146,188)(147,187)(148,186)(149,185)(150,184)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(157,170)(158,169)(159,168)(160,167)(191,232)(192,231)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)(211,221)(212,230)(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)(219,223)(220,222)>;
G:=Group( (1,116,56,77)(2,117,57,78)(3,118,58,79)(4,119,59,80)(5,120,60,71)(6,111,51,72)(7,112,52,73)(8,113,53,74)(9,114,54,75)(10,115,55,76)(11,90,30,94)(12,81,21,95)(13,82,22,96)(14,83,23,97)(15,84,24,98)(16,85,25,99)(17,86,26,100)(18,87,27,91)(19,88,28,92)(20,89,29,93)(31,169,224,190)(32,170,225,181)(33,161,226,182)(34,162,227,183)(35,163,228,184)(36,164,229,185)(37,165,230,186)(38,166,221,187)(39,167,222,188)(40,168,223,189)(41,122,62,101)(42,123,63,102)(43,124,64,103)(44,125,65,104)(45,126,66,105)(46,127,67,106)(47,128,68,107)(48,129,69,108)(49,130,70,109)(50,121,61,110)(131,196,180,235)(132,197,171,236)(133,198,172,237)(134,199,173,238)(135,200,174,239)(136,191,175,240)(137,192,176,231)(138,193,177,232)(139,194,178,233)(140,195,179,234)(141,206,155,220)(142,207,156,211)(143,208,157,212)(144,209,158,213)(145,210,159,214)(146,201,160,215)(147,202,151,216)(148,203,152,217)(149,204,153,218)(150,205,154,219), (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), (1,197,6,192)(2,196,7,191)(3,195,8,200)(4,194,9,199)(5,193,10,198)(11,35,16,40)(12,34,17,39)(13,33,18,38)(14,32,19,37)(15,31,20,36)(21,227,26,222)(22,226,27,221)(23,225,28,230)(24,224,29,229)(25,223,30,228)(41,207,46,202)(42,206,47,201)(43,205,48,210)(44,204,49,209)(45,203,50,208)(51,231,56,236)(52,240,57,235)(53,239,58,234)(54,238,59,233)(55,237,60,232)(61,212,66,217)(62,211,67,216)(63,220,68,215)(64,219,69,214)(65,218,70,213)(71,177,76,172)(72,176,77,171)(73,175,78,180)(74,174,79,179)(75,173,80,178)(81,183,86,188)(82,182,87,187)(83,181,88,186)(84,190,89,185)(85,189,90,184)(91,166,96,161)(92,165,97,170)(93,164,98,169)(94,163,99,168)(95,162,100,167)(101,156,106,151)(102,155,107,160)(103,154,108,159)(104,153,109,158)(105,152,110,157)(111,137,116,132)(112,136,117,131)(113,135,118,140)(114,134,119,139)(115,133,120,138)(121,143,126,148)(122,142,127,147)(123,141,128,146)(124,150,129,145)(125,149,130,144), (1,27,41)(2,28,42)(3,29,43)(4,30,44)(5,21,45)(6,22,46)(7,23,47)(8,24,48)(9,25,49)(10,26,50)(11,65,59)(12,66,60)(13,67,51)(14,68,52)(15,69,53)(16,70,54)(17,61,55)(18,62,56)(19,63,57)(20,64,58)(31,214,239)(32,215,240)(33,216,231)(34,217,232)(35,218,233)(36,219,234)(37,220,235)(38,211,236)(39,212,237)(40,213,238)(71,81,105)(72,82,106)(73,83,107)(74,84,108)(75,85,109)(76,86,110)(77,87,101)(78,88,102)(79,89,103)(80,90,104)(91,122,116)(92,123,117)(93,124,118)(94,125,119)(95,126,120)(96,127,111)(97,128,112)(98,129,113)(99,130,114)(100,121,115)(131,165,141)(132,166,142)(133,167,143)(134,168,144)(135,169,145)(136,170,146)(137,161,147)(138,162,148)(139,163,149)(140,164,150)(151,176,182)(152,177,183)(153,178,184)(154,179,185)(155,180,186)(156,171,187)(157,172,188)(158,173,189)(159,174,190)(160,175,181)(191,225,201)(192,226,202)(193,227,203)(194,228,204)(195,229,205)(196,230,206)(197,221,207)(198,222,208)(199,223,209)(200,224,210), (2,10)(3,9)(4,8)(5,7)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,70)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,50)(29,49)(30,48)(31,204)(32,203)(33,202)(34,201)(35,210)(36,209)(37,208)(38,207)(39,206)(40,205)(52,60)(53,59)(54,58)(55,57)(71,78)(72,77)(73,76)(74,75)(79,80)(81,102)(82,101)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,130)(99,129)(100,128)(111,116)(112,115)(113,114)(117,120)(118,119)(131,177)(132,176)(133,175)(134,174)(135,173)(136,172)(137,171)(138,180)(139,179)(140,178)(141,183)(142,182)(143,181)(144,190)(145,189)(146,188)(147,187)(148,186)(149,185)(150,184)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(157,170)(158,169)(159,168)(160,167)(191,232)(192,231)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)(211,221)(212,230)(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)(219,223)(220,222) );
G=PermutationGroup([[(1,116,56,77),(2,117,57,78),(3,118,58,79),(4,119,59,80),(5,120,60,71),(6,111,51,72),(7,112,52,73),(8,113,53,74),(9,114,54,75),(10,115,55,76),(11,90,30,94),(12,81,21,95),(13,82,22,96),(14,83,23,97),(15,84,24,98),(16,85,25,99),(17,86,26,100),(18,87,27,91),(19,88,28,92),(20,89,29,93),(31,169,224,190),(32,170,225,181),(33,161,226,182),(34,162,227,183),(35,163,228,184),(36,164,229,185),(37,165,230,186),(38,166,221,187),(39,167,222,188),(40,168,223,189),(41,122,62,101),(42,123,63,102),(43,124,64,103),(44,125,65,104),(45,126,66,105),(46,127,67,106),(47,128,68,107),(48,129,69,108),(49,130,70,109),(50,121,61,110),(131,196,180,235),(132,197,171,236),(133,198,172,237),(134,199,173,238),(135,200,174,239),(136,191,175,240),(137,192,176,231),(138,193,177,232),(139,194,178,233),(140,195,179,234),(141,206,155,220),(142,207,156,211),(143,208,157,212),(144,209,158,213),(145,210,159,214),(146,201,160,215),(147,202,151,216),(148,203,152,217),(149,204,153,218),(150,205,154,219)], [(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)], [(1,197,6,192),(2,196,7,191),(3,195,8,200),(4,194,9,199),(5,193,10,198),(11,35,16,40),(12,34,17,39),(13,33,18,38),(14,32,19,37),(15,31,20,36),(21,227,26,222),(22,226,27,221),(23,225,28,230),(24,224,29,229),(25,223,30,228),(41,207,46,202),(42,206,47,201),(43,205,48,210),(44,204,49,209),(45,203,50,208),(51,231,56,236),(52,240,57,235),(53,239,58,234),(54,238,59,233),(55,237,60,232),(61,212,66,217),(62,211,67,216),(63,220,68,215),(64,219,69,214),(65,218,70,213),(71,177,76,172),(72,176,77,171),(73,175,78,180),(74,174,79,179),(75,173,80,178),(81,183,86,188),(82,182,87,187),(83,181,88,186),(84,190,89,185),(85,189,90,184),(91,166,96,161),(92,165,97,170),(93,164,98,169),(94,163,99,168),(95,162,100,167),(101,156,106,151),(102,155,107,160),(103,154,108,159),(104,153,109,158),(105,152,110,157),(111,137,116,132),(112,136,117,131),(113,135,118,140),(114,134,119,139),(115,133,120,138),(121,143,126,148),(122,142,127,147),(123,141,128,146),(124,150,129,145),(125,149,130,144)], [(1,27,41),(2,28,42),(3,29,43),(4,30,44),(5,21,45),(6,22,46),(7,23,47),(8,24,48),(9,25,49),(10,26,50),(11,65,59),(12,66,60),(13,67,51),(14,68,52),(15,69,53),(16,70,54),(17,61,55),(18,62,56),(19,63,57),(20,64,58),(31,214,239),(32,215,240),(33,216,231),(34,217,232),(35,218,233),(36,219,234),(37,220,235),(38,211,236),(39,212,237),(40,213,238),(71,81,105),(72,82,106),(73,83,107),(74,84,108),(75,85,109),(76,86,110),(77,87,101),(78,88,102),(79,89,103),(80,90,104),(91,122,116),(92,123,117),(93,124,118),(94,125,119),(95,126,120),(96,127,111),(97,128,112),(98,129,113),(99,130,114),(100,121,115),(131,165,141),(132,166,142),(133,167,143),(134,168,144),(135,169,145),(136,170,146),(137,161,147),(138,162,148),(139,163,149),(140,164,150),(151,176,182),(152,177,183),(153,178,184),(154,179,185),(155,180,186),(156,171,187),(157,172,188),(158,173,189),(159,174,190),(160,175,181),(191,225,201),(192,226,202),(193,227,203),(194,228,204),(195,229,205),(196,230,206),(197,221,207),(198,222,208),(199,223,209),(200,224,210)], [(2,10),(3,9),(4,8),(5,7),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,70),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,50),(29,49),(30,48),(31,204),(32,203),(33,202),(34,201),(35,210),(36,209),(37,208),(38,207),(39,206),(40,205),(52,60),(53,59),(54,58),(55,57),(71,78),(72,77),(73,76),(74,75),(79,80),(81,102),(82,101),(83,110),(84,109),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(91,127),(92,126),(93,125),(94,124),(95,123),(96,122),(97,121),(98,130),(99,129),(100,128),(111,116),(112,115),(113,114),(117,120),(118,119),(131,177),(132,176),(133,175),(134,174),(135,173),(136,172),(137,171),(138,180),(139,179),(140,178),(141,183),(142,182),(143,181),(144,190),(145,189),(146,188),(147,187),(148,186),(149,185),(150,184),(151,166),(152,165),(153,164),(154,163),(155,162),(156,161),(157,170),(158,169),(159,168),(160,167),(191,232),(192,231),(193,240),(194,239),(195,238),(196,237),(197,236),(198,235),(199,234),(200,233),(211,221),(212,230),(213,229),(214,228),(215,227),(216,226),(217,225),(218,224),(219,223),(220,222)]])
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 | D20⋊S3 | D60⋊C2 | Dic3.D10 |
| kernel | C4⋊Dic5⋊S3 | Dic3×Dic5 | D30⋊4C4 | C30.Q8 | C3×C4⋊Dic5 | C5×Dic3⋊C4 | D30⋊3C4 | C4⋊Dic5 | Dic3⋊C4 | 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⋊Dic5⋊S3 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 57 | 0 | 0 |
| 0 | 0 | 58 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 3 | 50 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 16 | 17 | 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 | 39 | 8 | 0 | 0 |
| 0 | 0 | 8 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 54 |
| 0 | 0 | 0 | 0 | 56 | 24 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 60 | 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 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 18 | 0 | 0 |
| 0 | 0 | 45 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,32,58,0,0,0,0,57,29,0,0,0,0,0,0,11,3,0,0,0,0,0,50],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,1,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,39,8,0,0,0,0,8,22,0,0,0,0,0,0,37,56,0,0,0,0,54,24],[0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,1,28,0,0,0,0,0,60] >;
C4⋊Dic5⋊S3 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_5\rtimes S_3 % in TeX
G:=Group("C4:Dic5:S3"); // GroupNames label
G:=SmallGroup(480,421);
// by ID
G=gap.SmallGroup(480,421);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,590,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^10=d^3=e^2=1,c^2=b^5,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e=a*b^5,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