direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic3⋊4D4, C3⋊D4⋊C20, C3⋊2(D4×C20), C15⋊36(C4×D4), D6⋊C4⋊9C10, D6⋊2(C2×C20), C22⋊2(S3×C20), Dic3⋊4(C5×D4), C6.18(D4×C10), Dic3⋊C4⋊9C10, (C5×Dic3)⋊19D4, Dic3⋊1(C2×C20), C10.170(S3×D4), (C2×C20).271D6, C30.354(C2×D4), C6.7(C22×C20), (Dic3×C20)⋊29C2, (C4×Dic3)⋊11C10, C23.19(S3×C10), (C22×C10).89D6, C30.244(C4○D4), (C2×C30).401C23, C30.198(C22×C4), (C2×C60).415C22, (C22×Dic3)⋊1C10, C10.109(D4⋊2S3), (C22×C30).116C22, (C10×Dic3).215C22, C2.2(C5×S3×D4), (S3×C2×C4)⋊9C10, C2.9(S3×C2×C20), (S3×C2×C20)⋊25C2, (C2×C6)⋊2(C2×C20), (C5×C3⋊D4)⋊5C4, (C2×C30)⋊30(C2×C4), (C2×C10)⋊16(C4×S3), (C5×D6⋊C4)⋊31C2, C22⋊C4⋊7(C5×S3), C10.134(S3×C2×C4), C6.21(C5×C4○D4), (S3×C10)⋊24(C2×C4), (C3×C22⋊C4)⋊9C10, (C5×C22⋊C4)⋊15S3, (C2×C4).26(S3×C10), (Dic3×C2×C10)⋊12C2, C2.2(C5×D4⋊2S3), (C2×C3⋊D4).2C10, (C10×C3⋊D4).9C2, C22.14(S3×C2×C10), (C15×C22⋊C4)⋊25C2, (C2×C12).54(C2×C10), (C5×Dic3⋊C4)⋊31C2, (C5×Dic3)⋊17(C2×C4), (S3×C2×C10).107C22, (C2×C6).22(C22×C10), (C22×C6).11(C2×C10), (C22×S3).17(C2×C10), (C2×C10).335(C22×S3), (C2×Dic3).47(C2×C10), SmallGroup(480,760)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic3⋊4D4
G = < a,b,c,d,e | a5=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 404 in 188 conjugacy classes, 86 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C4×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Dic3⋊4D4, S3×C20, C10×Dic3, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, C22×C30, D4×C20, Dic3×C20, C5×Dic3⋊C4, C5×D6⋊C4, C15×C22⋊C4, S3×C2×C20, Dic3×C2×C10, C10×C3⋊D4, C5×Dic3⋊4D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C23, C10, D6, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C4×D4, C2×C20, C5×D4, C22×C10, S3×C2×C4, S3×D4, D4⋊2S3, S3×C10, C22×C20, D4×C10, C5×C4○D4, Dic3⋊4D4, S3×C20, S3×C2×C10, D4×C20, S3×C2×C20, C5×S3×D4, C5×D4⋊2S3, C5×Dic3⋊4D4
►On 240 points
Generators in S240
(1 58 46 34 22)(2 59 47 35 23)(3 60 48 36 24)(4 55 43 31 19)(5 56 44 32 20)(6 57 45 33 21)(7 231 219 207 195)(8 232 220 208 196)(9 233 221 209 197)(10 234 222 210 198)(11 229 217 205 193)(12 230 218 206 194)(13 61 49 37 25)(14 62 50 38 26)(15 63 51 39 27)(16 64 52 40 28)(17 65 53 41 29)(18 66 54 42 30)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(73 121 109 97 85)(74 122 110 98 86)(75 123 111 99 87)(76 124 112 100 88)(77 125 113 101 89)(78 126 114 102 90)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(133 181 169 157 145)(134 182 170 158 146)(135 183 171 159 147)(136 184 172 160 148)(137 185 173 161 149)(138 186 174 162 150)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(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 67 4 70)(2 72 5 69)(3 71 6 68)(7 181 10 184)(8 186 11 183)(9 185 12 182)(13 78 16 75)(14 77 17 74)(15 76 18 73)(19 82 22 79)(20 81 23 84)(21 80 24 83)(25 90 28 87)(26 89 29 86)(27 88 30 85)(31 94 34 91)(32 93 35 96)(33 92 36 95)(37 102 40 99)(38 101 41 98)(39 100 42 97)(43 106 46 103)(44 105 47 108)(45 104 48 107)(49 114 52 111)(50 113 53 110)(51 112 54 109)(55 118 58 115)(56 117 59 120)(57 116 60 119)(61 126 64 123)(62 125 65 122)(63 124 66 121)(127 190 130 187)(128 189 131 192)(129 188 132 191)(133 198 136 195)(134 197 137 194)(135 196 138 193)(139 202 142 199)(140 201 143 204)(141 200 144 203)(145 210 148 207)(146 209 149 206)(147 208 150 205)(151 214 154 211)(152 213 155 216)(153 212 156 215)(157 222 160 219)(158 221 161 218)(159 220 162 217)(163 226 166 223)(164 225 167 228)(165 224 168 227)(169 234 172 231)(170 233 173 230)(171 232 174 229)(175 238 178 235)(176 237 179 240)(177 236 180 239)
(1 137 17 130)(2 136 18 129)(3 135 13 128)(4 134 14 127)(5 133 15 132)(6 138 16 131)(7 121 236 120)(8 126 237 119)(9 125 238 118)(10 124 239 117)(11 123 240 116)(12 122 235 115)(19 146 26 139)(20 145 27 144)(21 150 28 143)(22 149 29 142)(23 148 30 141)(24 147 25 140)(31 158 38 151)(32 157 39 156)(33 162 40 155)(34 161 41 154)(35 160 42 153)(36 159 37 152)(43 170 50 163)(44 169 51 168)(45 174 52 167)(46 173 53 166)(47 172 54 165)(48 171 49 164)(55 182 62 175)(56 181 63 180)(57 186 64 179)(58 185 65 178)(59 184 66 177)(60 183 61 176)(67 194 74 187)(68 193 75 192)(69 198 76 191)(70 197 77 190)(71 196 78 189)(72 195 73 188)(79 206 86 199)(80 205 87 204)(81 210 88 203)(82 209 89 202)(83 208 90 201)(84 207 85 200)(91 218 98 211)(92 217 99 216)(93 222 100 215)(94 221 101 214)(95 220 102 213)(96 219 97 212)(103 230 110 223)(104 229 111 228)(105 234 112 227)(106 233 113 226)(107 232 114 225)(108 231 109 224)
(1 130)(2 131)(3 132)(4 127)(5 128)(6 129)(7 123)(8 124)(9 125)(10 126)(11 121)(12 122)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 174)(55 175)(56 176)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)
G:=sub<Sym(240)| (1,58,46,34,22)(2,59,47,35,23)(3,60,48,36,24)(4,55,43,31,19)(5,56,44,32,20)(6,57,45,33,21)(7,231,219,207,195)(8,232,220,208,196)(9,233,221,209,197)(10,234,222,210,198)(11,229,217,205,193)(12,230,218,206,194)(13,61,49,37,25)(14,62,50,38,26)(15,63,51,39,27)(16,64,52,40,28)(17,65,53,41,29)(18,66,54,42,30)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84)(73,121,109,97,85)(74,122,110,98,86)(75,123,111,99,87)(76,124,112,100,88)(77,125,113,101,89)(78,126,114,102,90)(127,175,163,151,139)(128,176,164,152,140)(129,177,165,153,141)(130,178,166,154,142)(131,179,167,155,143)(132,180,168,156,144)(133,181,169,157,145)(134,182,170,158,146)(135,183,171,159,147)(136,184,172,160,148)(137,185,173,161,149)(138,186,174,162,150)(187,235,223,211,199)(188,236,224,212,200)(189,237,225,213,201)(190,238,226,214,202)(191,239,227,215,203)(192,240,228,216,204), (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), (1,67,4,70)(2,72,5,69)(3,71,6,68)(7,181,10,184)(8,186,11,183)(9,185,12,182)(13,78,16,75)(14,77,17,74)(15,76,18,73)(19,82,22,79)(20,81,23,84)(21,80,24,83)(25,90,28,87)(26,89,29,86)(27,88,30,85)(31,94,34,91)(32,93,35,96)(33,92,36,95)(37,102,40,99)(38,101,41,98)(39,100,42,97)(43,106,46,103)(44,105,47,108)(45,104,48,107)(49,114,52,111)(50,113,53,110)(51,112,54,109)(55,118,58,115)(56,117,59,120)(57,116,60,119)(61,126,64,123)(62,125,65,122)(63,124,66,121)(127,190,130,187)(128,189,131,192)(129,188,132,191)(133,198,136,195)(134,197,137,194)(135,196,138,193)(139,202,142,199)(140,201,143,204)(141,200,144,203)(145,210,148,207)(146,209,149,206)(147,208,150,205)(151,214,154,211)(152,213,155,216)(153,212,156,215)(157,222,160,219)(158,221,161,218)(159,220,162,217)(163,226,166,223)(164,225,167,228)(165,224,168,227)(169,234,172,231)(170,233,173,230)(171,232,174,229)(175,238,178,235)(176,237,179,240)(177,236,180,239), (1,137,17,130)(2,136,18,129)(3,135,13,128)(4,134,14,127)(5,133,15,132)(6,138,16,131)(7,121,236,120)(8,126,237,119)(9,125,238,118)(10,124,239,117)(11,123,240,116)(12,122,235,115)(19,146,26,139)(20,145,27,144)(21,150,28,143)(22,149,29,142)(23,148,30,141)(24,147,25,140)(31,158,38,151)(32,157,39,156)(33,162,40,155)(34,161,41,154)(35,160,42,153)(36,159,37,152)(43,170,50,163)(44,169,51,168)(45,174,52,167)(46,173,53,166)(47,172,54,165)(48,171,49,164)(55,182,62,175)(56,181,63,180)(57,186,64,179)(58,185,65,178)(59,184,66,177)(60,183,61,176)(67,194,74,187)(68,193,75,192)(69,198,76,191)(70,197,77,190)(71,196,78,189)(72,195,73,188)(79,206,86,199)(80,205,87,204)(81,210,88,203)(82,209,89,202)(83,208,90,201)(84,207,85,200)(91,218,98,211)(92,217,99,216)(93,222,100,215)(94,221,101,214)(95,220,102,213)(96,219,97,212)(103,230,110,223)(104,229,111,228)(105,234,112,227)(106,233,113,226)(107,232,114,225)(108,231,109,224), (1,130)(2,131)(3,132)(4,127)(5,128)(6,129)(7,123)(8,124)(9,125)(10,126)(11,121)(12,122)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240)>;
G:=Group( (1,58,46,34,22)(2,59,47,35,23)(3,60,48,36,24)(4,55,43,31,19)(5,56,44,32,20)(6,57,45,33,21)(7,231,219,207,195)(8,232,220,208,196)(9,233,221,209,197)(10,234,222,210,198)(11,229,217,205,193)(12,230,218,206,194)(13,61,49,37,25)(14,62,50,38,26)(15,63,51,39,27)(16,64,52,40,28)(17,65,53,41,29)(18,66,54,42,30)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84)(73,121,109,97,85)(74,122,110,98,86)(75,123,111,99,87)(76,124,112,100,88)(77,125,113,101,89)(78,126,114,102,90)(127,175,163,151,139)(128,176,164,152,140)(129,177,165,153,141)(130,178,166,154,142)(131,179,167,155,143)(132,180,168,156,144)(133,181,169,157,145)(134,182,170,158,146)(135,183,171,159,147)(136,184,172,160,148)(137,185,173,161,149)(138,186,174,162,150)(187,235,223,211,199)(188,236,224,212,200)(189,237,225,213,201)(190,238,226,214,202)(191,239,227,215,203)(192,240,228,216,204), (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), (1,67,4,70)(2,72,5,69)(3,71,6,68)(7,181,10,184)(8,186,11,183)(9,185,12,182)(13,78,16,75)(14,77,17,74)(15,76,18,73)(19,82,22,79)(20,81,23,84)(21,80,24,83)(25,90,28,87)(26,89,29,86)(27,88,30,85)(31,94,34,91)(32,93,35,96)(33,92,36,95)(37,102,40,99)(38,101,41,98)(39,100,42,97)(43,106,46,103)(44,105,47,108)(45,104,48,107)(49,114,52,111)(50,113,53,110)(51,112,54,109)(55,118,58,115)(56,117,59,120)(57,116,60,119)(61,126,64,123)(62,125,65,122)(63,124,66,121)(127,190,130,187)(128,189,131,192)(129,188,132,191)(133,198,136,195)(134,197,137,194)(135,196,138,193)(139,202,142,199)(140,201,143,204)(141,200,144,203)(145,210,148,207)(146,209,149,206)(147,208,150,205)(151,214,154,211)(152,213,155,216)(153,212,156,215)(157,222,160,219)(158,221,161,218)(159,220,162,217)(163,226,166,223)(164,225,167,228)(165,224,168,227)(169,234,172,231)(170,233,173,230)(171,232,174,229)(175,238,178,235)(176,237,179,240)(177,236,180,239), (1,137,17,130)(2,136,18,129)(3,135,13,128)(4,134,14,127)(5,133,15,132)(6,138,16,131)(7,121,236,120)(8,126,237,119)(9,125,238,118)(10,124,239,117)(11,123,240,116)(12,122,235,115)(19,146,26,139)(20,145,27,144)(21,150,28,143)(22,149,29,142)(23,148,30,141)(24,147,25,140)(31,158,38,151)(32,157,39,156)(33,162,40,155)(34,161,41,154)(35,160,42,153)(36,159,37,152)(43,170,50,163)(44,169,51,168)(45,174,52,167)(46,173,53,166)(47,172,54,165)(48,171,49,164)(55,182,62,175)(56,181,63,180)(57,186,64,179)(58,185,65,178)(59,184,66,177)(60,183,61,176)(67,194,74,187)(68,193,75,192)(69,198,76,191)(70,197,77,190)(71,196,78,189)(72,195,73,188)(79,206,86,199)(80,205,87,204)(81,210,88,203)(82,209,89,202)(83,208,90,201)(84,207,85,200)(91,218,98,211)(92,217,99,216)(93,222,100,215)(94,221,101,214)(95,220,102,213)(96,219,97,212)(103,230,110,223)(104,229,111,228)(105,234,112,227)(106,233,113,226)(107,232,114,225)(108,231,109,224), (1,130)(2,131)(3,132)(4,127)(5,128)(6,129)(7,123)(8,124)(9,125)(10,126)(11,121)(12,122)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240) );
G=PermutationGroup([[(1,58,46,34,22),(2,59,47,35,23),(3,60,48,36,24),(4,55,43,31,19),(5,56,44,32,20),(6,57,45,33,21),(7,231,219,207,195),(8,232,220,208,196),(9,233,221,209,197),(10,234,222,210,198),(11,229,217,205,193),(12,230,218,206,194),(13,61,49,37,25),(14,62,50,38,26),(15,63,51,39,27),(16,64,52,40,28),(17,65,53,41,29),(18,66,54,42,30),(67,115,103,91,79),(68,116,104,92,80),(69,117,105,93,81),(70,118,106,94,82),(71,119,107,95,83),(72,120,108,96,84),(73,121,109,97,85),(74,122,110,98,86),(75,123,111,99,87),(76,124,112,100,88),(77,125,113,101,89),(78,126,114,102,90),(127,175,163,151,139),(128,176,164,152,140),(129,177,165,153,141),(130,178,166,154,142),(131,179,167,155,143),(132,180,168,156,144),(133,181,169,157,145),(134,182,170,158,146),(135,183,171,159,147),(136,184,172,160,148),(137,185,173,161,149),(138,186,174,162,150),(187,235,223,211,199),(188,236,224,212,200),(189,237,225,213,201),(190,238,226,214,202),(191,239,227,215,203),(192,240,228,216,204)], [(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)], [(1,67,4,70),(2,72,5,69),(3,71,6,68),(7,181,10,184),(8,186,11,183),(9,185,12,182),(13,78,16,75),(14,77,17,74),(15,76,18,73),(19,82,22,79),(20,81,23,84),(21,80,24,83),(25,90,28,87),(26,89,29,86),(27,88,30,85),(31,94,34,91),(32,93,35,96),(33,92,36,95),(37,102,40,99),(38,101,41,98),(39,100,42,97),(43,106,46,103),(44,105,47,108),(45,104,48,107),(49,114,52,111),(50,113,53,110),(51,112,54,109),(55,118,58,115),(56,117,59,120),(57,116,60,119),(61,126,64,123),(62,125,65,122),(63,124,66,121),(127,190,130,187),(128,189,131,192),(129,188,132,191),(133,198,136,195),(134,197,137,194),(135,196,138,193),(139,202,142,199),(140,201,143,204),(141,200,144,203),(145,210,148,207),(146,209,149,206),(147,208,150,205),(151,214,154,211),(152,213,155,216),(153,212,156,215),(157,222,160,219),(158,221,161,218),(159,220,162,217),(163,226,166,223),(164,225,167,228),(165,224,168,227),(169,234,172,231),(170,233,173,230),(171,232,174,229),(175,238,178,235),(176,237,179,240),(177,236,180,239)], [(1,137,17,130),(2,136,18,129),(3,135,13,128),(4,134,14,127),(5,133,15,132),(6,138,16,131),(7,121,236,120),(8,126,237,119),(9,125,238,118),(10,124,239,117),(11,123,240,116),(12,122,235,115),(19,146,26,139),(20,145,27,144),(21,150,28,143),(22,149,29,142),(23,148,30,141),(24,147,25,140),(31,158,38,151),(32,157,39,156),(33,162,40,155),(34,161,41,154),(35,160,42,153),(36,159,37,152),(43,170,50,163),(44,169,51,168),(45,174,52,167),(46,173,53,166),(47,172,54,165),(48,171,49,164),(55,182,62,175),(56,181,63,180),(57,186,64,179),(58,185,65,178),(59,184,66,177),(60,183,61,176),(67,194,74,187),(68,193,75,192),(69,198,76,191),(70,197,77,190),(71,196,78,189),(72,195,73,188),(79,206,86,199),(80,205,87,204),(81,210,88,203),(82,209,89,202),(83,208,90,201),(84,207,85,200),(91,218,98,211),(92,217,99,216),(93,222,100,215),(94,221,101,214),(95,220,102,213),(96,219,97,212),(103,230,110,223),(104,229,111,228),(105,234,112,227),(106,233,113,226),(107,232,114,225),(108,231,109,224)], [(1,130),(2,131),(3,132),(4,127),(5,128),(6,129),(7,123),(8,124),(9,125),(10,126),(11,121),(12,122),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,141),(22,142),(23,143),(24,144),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(41,161),(42,162),(43,163),(44,164),(45,165),(46,166),(47,167),(48,168),(49,169),(50,170),(51,171),(52,172),(53,173),(54,174),(55,175),(56,176),(57,177),(58,178),(59,179),(60,180),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190),(71,191),(72,192),(73,193),(74,194),(75,195),(76,196),(77,197),(78,198),(79,199),(80,200),(81,201),(82,202),(83,203),(84,204),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,225),(106,226),(107,227),(108,228),(109,229),(110,230),(111,231),(112,232),(113,233),(114,234),(115,235),(116,236),(117,237),(118,238),(119,239),(120,240)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AV | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C20 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | C5×S3 | C5×D4 | S3×C10 | S3×C10 | C5×C4○D4 | S3×C20 | S3×D4 | D4⋊2S3 | C5×S3×D4 | C5×D4⋊2S3 |
| kernel | C5×Dic3⋊4D4 | Dic3×C20 | C5×Dic3⋊C4 | C5×D6⋊C4 | C15×C22⋊C4 | S3×C2×C20 | Dic3×C2×C10 | C10×C3⋊D4 | C5×C3⋊D4 | Dic3⋊4D4 | C4×Dic3 | Dic3⋊C4 | D6⋊C4 | C3×C22⋊C4 | S3×C2×C4 | C22×Dic3 | C2×C3⋊D4 | C3⋊D4 | C5×C22⋊C4 | C5×Dic3 | C2×C20 | C22×C10 | C30 | C2×C10 | C22⋊C4 | Dic3 | C2×C4 | C23 | C6 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 32 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×Dic3⋊4D4 ►in GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 1 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 11 | 0 | 0 | 0 |
| 50 | 50 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 1 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 11 | 3 |
| 0 | 0 | 0 | 50 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 3 |
| 0 | 0 | 21 | 50 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,60,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[11,50,0,0,0,50,0,0,0,0,50,0,0,0,0,50],[1,60,0,0,0,60,0,0,0,0,11,0,0,0,3,50],[1,0,0,0,0,1,0,0,0,0,11,21,0,0,3,50] >;
C5×Dic3⋊4D4 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_3\rtimes_4D_4 % in TeX
G:=Group("C5xDic3:4D4"); // GroupNames label
G:=SmallGroup(480,760);
// by ID
G=gap.SmallGroup(480,760);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,891,226,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations