direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic3⋊D4, D6⋊1(C5×D4), D6⋊C4⋊10C10, (S3×C10)⋊13D4, (C2×D12)⋊3C10, Dic3⋊2(C5×D4), C6.20(D4×C10), (C10×D12)⋊19C2, C15⋊27(C4⋊D4), Dic3⋊C4⋊5C10, (C5×Dic3)⋊17D4, C10.173(S3×D4), (C2×C20).234D6, C30.356(C2×D4), C23.10(S3×C10), (C22×C10).22D6, C30.204(C4○D4), (C2×C60).328C22, (C2×C30).404C23, C10.116(C4○D12), (C22×C30).119C22, (C10×Dic3).140C22, C2.9(C5×S3×D4), C3⋊1(C5×C4⋊D4), (S3×C2×C4)⋊11C10, (S3×C2×C20)⋊27C2, C6.8(C5×C4○D4), (C5×D6⋊C4)⋊32C2, (C2×C3⋊D4)⋊2C10, C22⋊C4⋊4(C5×S3), (C10×C3⋊D4)⋊17C2, (C5×C22⋊C4)⋊12S3, (C3×C22⋊C4)⋊6C10, (C2×C4).27(S3×C10), C2.11(C5×C4○D12), C22.43(S3×C2×C10), (C15×C22⋊C4)⋊20C2, (C2×C12).55(C2×C10), (C5×Dic3⋊C4)⋊21C2, (C22×S3).4(C2×C10), (S3×C2×C10).109C22, (C22×C6).14(C2×C10), (C2×C6).25(C22×C10), (C2×C10).338(C22×S3), (C2×Dic3).22(C2×C10), SmallGroup(480,763)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic3⋊D4
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=ebe=b-1, dcd-1=b3c, ce=ec, ede=d-1 >
Subgroups: 500 in 188 conjugacy classes, 66 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C4×S3, D12, 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, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C5×Dic3, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Dic3⋊D4, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, C22×C30, C5×C4⋊D4, C5×Dic3⋊C4, C5×D6⋊C4, C15×C22⋊C4, S3×C2×C20, C10×D12, C10×C3⋊D4, C5×Dic3⋊D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, C22×S3, C5×S3, C4⋊D4, C5×D4, C22×C10, C4○D12, S3×D4, S3×C10, D4×C10, C5×C4○D4, Dic3⋊D4, S3×C2×C10, C5×C4⋊D4, C5×C4○D12, C5×S3×D4, C5×Dic3⋊D4
►On 240 points
Generators in S240
(1 60 48 36 24)(2 55 43 31 19)(3 56 44 32 20)(4 57 45 33 21)(5 58 46 34 22)(6 59 47 35 23)(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)
(1 134 4 137)(2 133 5 136)(3 138 6 135)(7 124 10 121)(8 123 11 126)(9 122 12 125)(13 127 16 130)(14 132 17 129)(15 131 18 128)(19 145 22 148)(20 150 23 147)(21 149 24 146)(25 139 28 142)(26 144 29 141)(27 143 30 140)(31 157 34 160)(32 162 35 159)(33 161 36 158)(37 151 40 154)(38 156 41 153)(39 155 42 152)(43 169 46 172)(44 174 47 171)(45 173 48 170)(49 163 52 166)(50 168 53 165)(51 167 54 164)(55 181 58 184)(56 186 59 183)(57 185 60 182)(61 175 64 178)(62 180 65 177)(63 179 66 176)(67 191 70 188)(68 190 71 187)(69 189 72 192)(73 195 76 198)(74 194 77 197)(75 193 78 196)(79 203 82 200)(80 202 83 199)(81 201 84 204)(85 207 88 210)(86 206 89 209)(87 205 90 208)(91 215 94 212)(92 214 95 211)(93 213 96 216)(97 219 100 222)(98 218 101 221)(99 217 102 220)(103 227 106 224)(104 226 107 223)(105 225 108 228)(109 231 112 234)(110 230 113 233)(111 229 114 232)(115 239 118 236)(116 238 119 235)(117 237 120 240)
(1 77 14 69)(2 76 15 68)(3 75 16 67)(4 74 17 72)(5 73 18 71)(6 78 13 70)(7 179 235 181)(8 178 236 186)(9 177 237 185)(10 176 238 184)(11 175 239 183)(12 180 240 182)(19 88 27 80)(20 87 28 79)(21 86 29 84)(22 85 30 83)(23 90 25 82)(24 89 26 81)(31 100 39 92)(32 99 40 91)(33 98 41 96)(34 97 42 95)(35 102 37 94)(36 101 38 93)(43 112 51 104)(44 111 52 103)(45 110 53 108)(46 109 54 107)(47 114 49 106)(48 113 50 105)(55 124 63 116)(56 123 64 115)(57 122 65 120)(58 121 66 119)(59 126 61 118)(60 125 62 117)(127 191 135 193)(128 190 136 198)(129 189 137 197)(130 188 138 196)(131 187 133 195)(132 192 134 194)(139 203 147 205)(140 202 148 210)(141 201 149 209)(142 200 150 208)(143 199 145 207)(144 204 146 206)(151 215 159 217)(152 214 160 222)(153 213 161 221)(154 212 162 220)(155 211 157 219)(156 216 158 218)(163 227 171 229)(164 226 172 234)(165 225 173 233)(166 224 174 232)(167 223 169 231)(168 228 170 230)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 18)(7 8)(9 12)(10 11)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 40)(32 39)(33 38)(34 37)(35 42)(36 41)(43 52)(44 51)(45 50)(46 49)(47 54)(48 53)(55 64)(56 63)(57 62)(58 61)(59 66)(60 65)(67 68)(69 72)(70 71)(73 78)(74 77)(75 76)(79 80)(81 84)(82 83)(85 90)(86 89)(87 88)(91 92)(93 96)(94 95)(97 102)(98 101)(99 100)(103 104)(105 108)(106 107)(109 114)(110 113)(111 112)(115 116)(117 120)(118 119)(121 126)(122 125)(123 124)(127 136)(128 135)(129 134)(130 133)(131 138)(132 137)(139 148)(140 147)(141 146)(142 145)(143 150)(144 149)(151 160)(152 159)(153 158)(154 157)(155 162)(156 161)(163 172)(164 171)(165 170)(166 169)(167 174)(168 173)(175 184)(176 183)(177 182)(178 181)(179 186)(180 185)(187 188)(189 192)(190 191)(193 198)(194 197)(195 196)(199 200)(201 204)(202 203)(205 210)(206 209)(207 208)(211 212)(213 216)(214 215)(217 222)(218 221)(219 220)(223 224)(225 228)(226 227)(229 234)(230 233)(231 232)(235 236)(237 240)(238 239)
G:=sub<Sym(240)| (1,60,48,36,24)(2,55,43,31,19)(3,56,44,32,20)(4,57,45,33,21)(5,58,46,34,22)(6,59,47,35,23)(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), (1,134,4,137)(2,133,5,136)(3,138,6,135)(7,124,10,121)(8,123,11,126)(9,122,12,125)(13,127,16,130)(14,132,17,129)(15,131,18,128)(19,145,22,148)(20,150,23,147)(21,149,24,146)(25,139,28,142)(26,144,29,141)(27,143,30,140)(31,157,34,160)(32,162,35,159)(33,161,36,158)(37,151,40,154)(38,156,41,153)(39,155,42,152)(43,169,46,172)(44,174,47,171)(45,173,48,170)(49,163,52,166)(50,168,53,165)(51,167,54,164)(55,181,58,184)(56,186,59,183)(57,185,60,182)(61,175,64,178)(62,180,65,177)(63,179,66,176)(67,191,70,188)(68,190,71,187)(69,189,72,192)(73,195,76,198)(74,194,77,197)(75,193,78,196)(79,203,82,200)(80,202,83,199)(81,201,84,204)(85,207,88,210)(86,206,89,209)(87,205,90,208)(91,215,94,212)(92,214,95,211)(93,213,96,216)(97,219,100,222)(98,218,101,221)(99,217,102,220)(103,227,106,224)(104,226,107,223)(105,225,108,228)(109,231,112,234)(110,230,113,233)(111,229,114,232)(115,239,118,236)(116,238,119,235)(117,237,120,240), (1,77,14,69)(2,76,15,68)(3,75,16,67)(4,74,17,72)(5,73,18,71)(6,78,13,70)(7,179,235,181)(8,178,236,186)(9,177,237,185)(10,176,238,184)(11,175,239,183)(12,180,240,182)(19,88,27,80)(20,87,28,79)(21,86,29,84)(22,85,30,83)(23,90,25,82)(24,89,26,81)(31,100,39,92)(32,99,40,91)(33,98,41,96)(34,97,42,95)(35,102,37,94)(36,101,38,93)(43,112,51,104)(44,111,52,103)(45,110,53,108)(46,109,54,107)(47,114,49,106)(48,113,50,105)(55,124,63,116)(56,123,64,115)(57,122,65,120)(58,121,66,119)(59,126,61,118)(60,125,62,117)(127,191,135,193)(128,190,136,198)(129,189,137,197)(130,188,138,196)(131,187,133,195)(132,192,134,194)(139,203,147,205)(140,202,148,210)(141,201,149,209)(142,200,150,208)(143,199,145,207)(144,204,146,206)(151,215,159,217)(152,214,160,222)(153,213,161,221)(154,212,162,220)(155,211,157,219)(156,216,158,218)(163,227,171,229)(164,226,172,234)(165,225,173,233)(166,224,174,232)(167,223,169,231)(168,228,170,230), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(7,8)(9,12)(10,11)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,40)(32,39)(33,38)(34,37)(35,42)(36,41)(43,52)(44,51)(45,50)(46,49)(47,54)(48,53)(55,64)(56,63)(57,62)(58,61)(59,66)(60,65)(67,68)(69,72)(70,71)(73,78)(74,77)(75,76)(79,80)(81,84)(82,83)(85,90)(86,89)(87,88)(91,92)(93,96)(94,95)(97,102)(98,101)(99,100)(103,104)(105,108)(106,107)(109,114)(110,113)(111,112)(115,116)(117,120)(118,119)(121,126)(122,125)(123,124)(127,136)(128,135)(129,134)(130,133)(131,138)(132,137)(139,148)(140,147)(141,146)(142,145)(143,150)(144,149)(151,160)(152,159)(153,158)(154,157)(155,162)(156,161)(163,172)(164,171)(165,170)(166,169)(167,174)(168,173)(175,184)(176,183)(177,182)(178,181)(179,186)(180,185)(187,188)(189,192)(190,191)(193,198)(194,197)(195,196)(199,200)(201,204)(202,203)(205,210)(206,209)(207,208)(211,212)(213,216)(214,215)(217,222)(218,221)(219,220)(223,224)(225,228)(226,227)(229,234)(230,233)(231,232)(235,236)(237,240)(238,239)>;
G:=Group( (1,60,48,36,24)(2,55,43,31,19)(3,56,44,32,20)(4,57,45,33,21)(5,58,46,34,22)(6,59,47,35,23)(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), (1,134,4,137)(2,133,5,136)(3,138,6,135)(7,124,10,121)(8,123,11,126)(9,122,12,125)(13,127,16,130)(14,132,17,129)(15,131,18,128)(19,145,22,148)(20,150,23,147)(21,149,24,146)(25,139,28,142)(26,144,29,141)(27,143,30,140)(31,157,34,160)(32,162,35,159)(33,161,36,158)(37,151,40,154)(38,156,41,153)(39,155,42,152)(43,169,46,172)(44,174,47,171)(45,173,48,170)(49,163,52,166)(50,168,53,165)(51,167,54,164)(55,181,58,184)(56,186,59,183)(57,185,60,182)(61,175,64,178)(62,180,65,177)(63,179,66,176)(67,191,70,188)(68,190,71,187)(69,189,72,192)(73,195,76,198)(74,194,77,197)(75,193,78,196)(79,203,82,200)(80,202,83,199)(81,201,84,204)(85,207,88,210)(86,206,89,209)(87,205,90,208)(91,215,94,212)(92,214,95,211)(93,213,96,216)(97,219,100,222)(98,218,101,221)(99,217,102,220)(103,227,106,224)(104,226,107,223)(105,225,108,228)(109,231,112,234)(110,230,113,233)(111,229,114,232)(115,239,118,236)(116,238,119,235)(117,237,120,240), (1,77,14,69)(2,76,15,68)(3,75,16,67)(4,74,17,72)(5,73,18,71)(6,78,13,70)(7,179,235,181)(8,178,236,186)(9,177,237,185)(10,176,238,184)(11,175,239,183)(12,180,240,182)(19,88,27,80)(20,87,28,79)(21,86,29,84)(22,85,30,83)(23,90,25,82)(24,89,26,81)(31,100,39,92)(32,99,40,91)(33,98,41,96)(34,97,42,95)(35,102,37,94)(36,101,38,93)(43,112,51,104)(44,111,52,103)(45,110,53,108)(46,109,54,107)(47,114,49,106)(48,113,50,105)(55,124,63,116)(56,123,64,115)(57,122,65,120)(58,121,66,119)(59,126,61,118)(60,125,62,117)(127,191,135,193)(128,190,136,198)(129,189,137,197)(130,188,138,196)(131,187,133,195)(132,192,134,194)(139,203,147,205)(140,202,148,210)(141,201,149,209)(142,200,150,208)(143,199,145,207)(144,204,146,206)(151,215,159,217)(152,214,160,222)(153,213,161,221)(154,212,162,220)(155,211,157,219)(156,216,158,218)(163,227,171,229)(164,226,172,234)(165,225,173,233)(166,224,174,232)(167,223,169,231)(168,228,170,230), (1,17)(2,16)(3,15)(4,14)(5,13)(6,18)(7,8)(9,12)(10,11)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,40)(32,39)(33,38)(34,37)(35,42)(36,41)(43,52)(44,51)(45,50)(46,49)(47,54)(48,53)(55,64)(56,63)(57,62)(58,61)(59,66)(60,65)(67,68)(69,72)(70,71)(73,78)(74,77)(75,76)(79,80)(81,84)(82,83)(85,90)(86,89)(87,88)(91,92)(93,96)(94,95)(97,102)(98,101)(99,100)(103,104)(105,108)(106,107)(109,114)(110,113)(111,112)(115,116)(117,120)(118,119)(121,126)(122,125)(123,124)(127,136)(128,135)(129,134)(130,133)(131,138)(132,137)(139,148)(140,147)(141,146)(142,145)(143,150)(144,149)(151,160)(152,159)(153,158)(154,157)(155,162)(156,161)(163,172)(164,171)(165,170)(166,169)(167,174)(168,173)(175,184)(176,183)(177,182)(178,181)(179,186)(180,185)(187,188)(189,192)(190,191)(193,198)(194,197)(195,196)(199,200)(201,204)(202,203)(205,210)(206,209)(207,208)(211,212)(213,216)(214,215)(217,222)(218,221)(219,220)(223,224)(225,228)(226,227)(229,234)(230,233)(231,232)(235,236)(237,240)(238,239) );
G=PermutationGroup([[(1,60,48,36,24),(2,55,43,31,19),(3,56,44,32,20),(4,57,45,33,21),(5,58,46,34,22),(6,59,47,35,23),(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)], [(1,134,4,137),(2,133,5,136),(3,138,6,135),(7,124,10,121),(8,123,11,126),(9,122,12,125),(13,127,16,130),(14,132,17,129),(15,131,18,128),(19,145,22,148),(20,150,23,147),(21,149,24,146),(25,139,28,142),(26,144,29,141),(27,143,30,140),(31,157,34,160),(32,162,35,159),(33,161,36,158),(37,151,40,154),(38,156,41,153),(39,155,42,152),(43,169,46,172),(44,174,47,171),(45,173,48,170),(49,163,52,166),(50,168,53,165),(51,167,54,164),(55,181,58,184),(56,186,59,183),(57,185,60,182),(61,175,64,178),(62,180,65,177),(63,179,66,176),(67,191,70,188),(68,190,71,187),(69,189,72,192),(73,195,76,198),(74,194,77,197),(75,193,78,196),(79,203,82,200),(80,202,83,199),(81,201,84,204),(85,207,88,210),(86,206,89,209),(87,205,90,208),(91,215,94,212),(92,214,95,211),(93,213,96,216),(97,219,100,222),(98,218,101,221),(99,217,102,220),(103,227,106,224),(104,226,107,223),(105,225,108,228),(109,231,112,234),(110,230,113,233),(111,229,114,232),(115,239,118,236),(116,238,119,235),(117,237,120,240)], [(1,77,14,69),(2,76,15,68),(3,75,16,67),(4,74,17,72),(5,73,18,71),(6,78,13,70),(7,179,235,181),(8,178,236,186),(9,177,237,185),(10,176,238,184),(11,175,239,183),(12,180,240,182),(19,88,27,80),(20,87,28,79),(21,86,29,84),(22,85,30,83),(23,90,25,82),(24,89,26,81),(31,100,39,92),(32,99,40,91),(33,98,41,96),(34,97,42,95),(35,102,37,94),(36,101,38,93),(43,112,51,104),(44,111,52,103),(45,110,53,108),(46,109,54,107),(47,114,49,106),(48,113,50,105),(55,124,63,116),(56,123,64,115),(57,122,65,120),(58,121,66,119),(59,126,61,118),(60,125,62,117),(127,191,135,193),(128,190,136,198),(129,189,137,197),(130,188,138,196),(131,187,133,195),(132,192,134,194),(139,203,147,205),(140,202,148,210),(141,201,149,209),(142,200,150,208),(143,199,145,207),(144,204,146,206),(151,215,159,217),(152,214,160,222),(153,213,161,221),(154,212,162,220),(155,211,157,219),(156,216,158,218),(163,227,171,229),(164,226,172,234),(165,225,173,233),(166,224,174,232),(167,223,169,231),(168,228,170,230)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,18),(7,8),(9,12),(10,11),(19,28),(20,27),(21,26),(22,25),(23,30),(24,29),(31,40),(32,39),(33,38),(34,37),(35,42),(36,41),(43,52),(44,51),(45,50),(46,49),(47,54),(48,53),(55,64),(56,63),(57,62),(58,61),(59,66),(60,65),(67,68),(69,72),(70,71),(73,78),(74,77),(75,76),(79,80),(81,84),(82,83),(85,90),(86,89),(87,88),(91,92),(93,96),(94,95),(97,102),(98,101),(99,100),(103,104),(105,108),(106,107),(109,114),(110,113),(111,112),(115,116),(117,120),(118,119),(121,126),(122,125),(123,124),(127,136),(128,135),(129,134),(130,133),(131,138),(132,137),(139,148),(140,147),(141,146),(142,145),(143,150),(144,149),(151,160),(152,159),(153,158),(154,157),(155,162),(156,161),(163,172),(164,171),(165,170),(166,169),(167,174),(168,173),(175,184),(176,183),(177,182),(178,181),(179,186),(180,185),(187,188),(189,192),(190,191),(193,198),(194,197),(195,196),(199,200),(201,204),(202,203),(205,210),(206,209),(207,208),(211,212),(213,216),(214,215),(217,222),(218,221),(219,220),(223,224),(225,228),(226,227),(229,234),(230,233),(231,232),(235,236),(237,240),(238,239)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 10Q | ··· | 10X | 10Y | 10Z | 10AA | 10AB | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 20U | 20V | 20W | 20X | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C5×S3 | C5×D4 | C5×D4 | C4○D12 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C4○D12 | S3×D4 | C5×S3×D4 |
| kernel | C5×Dic3⋊D4 | C5×Dic3⋊C4 | C5×D6⋊C4 | C15×C22⋊C4 | S3×C2×C20 | C10×D12 | C10×C3⋊D4 | Dic3⋊D4 | Dic3⋊C4 | D6⋊C4 | C3×C22⋊C4 | S3×C2×C4 | C2×D12 | C2×C3⋊D4 | C5×C22⋊C4 | C5×Dic3 | S3×C10 | C2×C20 | C22×C10 | C30 | C22⋊C4 | Dic3 | D6 | C10 | C2×C4 | C23 | C6 | C2 | C10 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 8 | 8 | 4 | 8 | 4 | 8 | 16 | 2 | 8 |
Matrix representation of C5×Dic3⋊D4 ►in GL6(𝔽61)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 0 | 0 | 34 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 19 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 58 |
| 0 | 0 | 0 | 0 | 0 | 50 |
| 29 | 21 | 0 | 0 | 0 | 0 |
| 18 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 2 |
| 0 | 0 | 0 | 0 | 3 | 19 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 35 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,0,0,0,0,0,0,34],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[11,19,0,0,0,0,0,50,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,58,50],[29,18,0,0,0,0,21,32,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,42,3,0,0,0,0,2,19],[1,35,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
C5×Dic3⋊D4 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_3\rtimes D_4 % in TeX
G:=Group("C5xDic3:D4"); // GroupNames label
G:=SmallGroup(480,763);
// by ID
G=gap.SmallGroup(480,763);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,288,2606,891,15686]);
// 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=e*b*e=b^-1,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations