metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C4.7D5, C4⋊Dic5⋊2S3, (C2×C20).6D6, D6⋊Dic5.2C2, C30.Q8⋊1C2, (C2×Dic5).4D6, C15⋊2(C42⋊2C2), (Dic3×Dic5)⋊4C2, C6.47(C4○D20), C30.10(C4○D4), (C2×C12).218D10, (C2×C30).31C23, C30.4Q8⋊12C2, (C22×S3).2D10, C10.66(C4○D12), C2.9(D12⋊D5), C6.35(D4⋊2D5), C2.7(D20⋊5S3), (C2×C60).249C22, (C2×Dic3).81D10, C3⋊4(C23.D10), C10.20(D4⋊2S3), C10.23(Q8⋊3S3), C2.9(Dic3.D10), (C6×Dic5).16C22, (C10×Dic3).15C22, (C2×Dic15).38C22, C5⋊4(C4⋊C4⋊S3), (C5×D6⋊C4).7C2, (C2×C4).28(S3×D5), (C3×C4⋊Dic5)⋊14C2, (S3×C2×C10).2C22, C22.122(C2×S3×D5), (C2×C6).43(C22×D5), (C2×C10).43(C22×S3), SmallGroup(480,417)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C4.D5
G = < a,b,c,d,e | a6=b2=c4=d5=1, e2=a3, bab=a-1, ac=ca, ad=da, ae=ea, cbc-1=a3b, bd=db, ebe-1=bc2, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 524 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C4⋊C4⋊S3, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C23.D10, Dic3×Dic5, D6⋊Dic5, C30.Q8, C3×C4⋊Dic5, C5×D6⋊C4, C30.4Q8, D6⋊C4.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, C4⋊C4⋊S3, C2×S3×D5, C23.D10, D20⋊5S3, D12⋊D5, Dic3.D10, D6⋊C4.D5
►On 240 points
Generators in S240
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 11)(8 10)(13 17)(14 16)(19 23)(20 22)(25 27)(28 30)(31 63)(32 62)(33 61)(34 66)(35 65)(36 64)(37 67)(38 72)(39 71)(40 70)(41 69)(42 68)(43 73)(44 78)(45 77)(46 76)(47 75)(48 74)(49 79)(50 84)(51 83)(52 82)(53 81)(54 80)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 130)(98 129)(99 128)(100 127)(101 132)(102 131)(103 136)(104 135)(105 134)(106 133)(107 138)(108 137)(109 142)(110 141)(111 140)(112 139)(113 144)(114 143)(145 150)(146 149)(147 148)(151 156)(152 155)(153 154)(157 160)(158 159)(161 162)(163 166)(164 165)(167 168)(169 172)(170 171)(173 174)(175 180)(176 179)(177 178)(181 186)(182 185)(183 184)(187 190)(188 189)(191 192)(193 196)(194 195)(197 198)(199 202)(200 201)(203 204)(205 207)(208 210)(211 213)(214 216)(218 222)(219 221)(224 228)(225 227)(230 234)(231 233)(235 237)(238 240)
(1 115 55 88)(2 116 56 89)(3 117 57 90)(4 118 58 85)(5 119 59 86)(6 120 60 87)(7 164 224 197)(8 165 225 198)(9 166 226 193)(10 167 227 194)(11 168 228 195)(12 163 223 196)(13 170 230 203)(14 171 231 204)(15 172 232 199)(16 173 233 200)(17 174 234 201)(18 169 229 202)(19 156 216 183)(20 151 211 184)(21 152 212 185)(22 153 213 186)(23 154 214 181)(24 155 215 182)(25 162 222 189)(26 157 217 190)(27 158 218 191)(28 159 219 192)(29 160 220 187)(30 161 221 188)(31 124 64 91)(32 125 65 92)(33 126 66 93)(34 121 61 94)(35 122 62 95)(36 123 63 96)(37 130 70 97)(38 131 71 98)(39 132 72 99)(40 127 67 100)(41 128 68 101)(42 129 69 102)(43 136 76 103)(44 137 77 104)(45 138 78 105)(46 133 73 106)(47 134 74 107)(48 135 75 108)(49 142 82 109)(50 143 83 110)(51 144 84 111)(52 139 79 112)(53 140 80 113)(54 141 81 114)(145 205 178 238)(146 206 179 239)(147 207 180 240)(148 208 175 235)(149 209 176 236)(150 210 177 237)
(1 39 51 48 31)(2 40 52 43 32)(3 41 53 44 33)(4 42 54 45 34)(5 37 49 46 35)(6 38 50 47 36)(7 16 30 237 22)(8 17 25 238 23)(9 18 26 239 24)(10 13 27 240 19)(11 14 28 235 20)(12 15 29 236 21)(55 72 84 75 64)(56 67 79 76 65)(57 68 80 77 66)(58 69 81 78 61)(59 70 82 73 62)(60 71 83 74 63)(85 102 114 105 94)(86 97 109 106 95)(87 98 110 107 96)(88 99 111 108 91)(89 100 112 103 92)(90 101 113 104 93)(115 132 144 135 124)(116 127 139 136 125)(117 128 140 137 126)(118 129 141 138 121)(119 130 142 133 122)(120 131 143 134 123)(145 154 165 174 162)(146 155 166 169 157)(147 156 167 170 158)(148 151 168 171 159)(149 152 163 172 160)(150 153 164 173 161)(175 184 195 204 192)(176 185 196 199 187)(177 186 197 200 188)(178 181 198 201 189)(179 182 193 202 190)(180 183 194 203 191)(205 214 225 234 222)(206 215 226 229 217)(207 216 227 230 218)(208 211 228 231 219)(209 212 223 232 220)(210 213 224 233 221)
(1 162 4 159)(2 157 5 160)(3 158 6 161)(7 140 10 143)(8 141 11 144)(9 142 12 139)(13 131 16 128)(14 132 17 129)(15 127 18 130)(19 134 22 137)(20 135 23 138)(21 136 24 133)(25 118 28 115)(26 119 29 116)(27 120 30 117)(31 145 34 148)(32 146 35 149)(33 147 36 150)(37 172 40 169)(38 173 41 170)(39 174 42 171)(43 155 46 152)(44 156 47 153)(45 151 48 154)(49 163 52 166)(50 164 53 167)(51 165 54 168)(55 189 58 192)(56 190 59 187)(57 191 60 188)(61 175 64 178)(62 176 65 179)(63 177 66 180)(67 202 70 199)(68 203 71 200)(69 204 72 201)(73 185 76 182)(74 186 77 183)(75 181 78 184)(79 193 82 196)(80 194 83 197)(81 195 84 198)(85 219 88 222)(86 220 89 217)(87 221 90 218)(91 205 94 208)(92 206 95 209)(93 207 96 210)(97 232 100 229)(98 233 101 230)(99 234 102 231)(103 215 106 212)(104 216 107 213)(105 211 108 214)(109 223 112 226)(110 224 113 227)(111 225 114 228)(121 235 124 238)(122 236 125 239)(123 237 126 240)
G:=sub<Sym(240)| (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,11)(8,10)(13,17)(14,16)(19,23)(20,22)(25,27)(28,30)(31,63)(32,62)(33,61)(34,66)(35,65)(36,64)(37,67)(38,72)(39,71)(40,70)(41,69)(42,68)(43,73)(44,78)(45,77)(46,76)(47,75)(48,74)(49,79)(50,84)(51,83)(52,82)(53,81)(54,80)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,130)(98,129)(99,128)(100,127)(101,132)(102,131)(103,136)(104,135)(105,134)(106,133)(107,138)(108,137)(109,142)(110,141)(111,140)(112,139)(113,144)(114,143)(145,150)(146,149)(147,148)(151,156)(152,155)(153,154)(157,160)(158,159)(161,162)(163,166)(164,165)(167,168)(169,172)(170,171)(173,174)(175,180)(176,179)(177,178)(181,186)(182,185)(183,184)(187,190)(188,189)(191,192)(193,196)(194,195)(197,198)(199,202)(200,201)(203,204)(205,207)(208,210)(211,213)(214,216)(218,222)(219,221)(224,228)(225,227)(230,234)(231,233)(235,237)(238,240), (1,115,55,88)(2,116,56,89)(3,117,57,90)(4,118,58,85)(5,119,59,86)(6,120,60,87)(7,164,224,197)(8,165,225,198)(9,166,226,193)(10,167,227,194)(11,168,228,195)(12,163,223,196)(13,170,230,203)(14,171,231,204)(15,172,232,199)(16,173,233,200)(17,174,234,201)(18,169,229,202)(19,156,216,183)(20,151,211,184)(21,152,212,185)(22,153,213,186)(23,154,214,181)(24,155,215,182)(25,162,222,189)(26,157,217,190)(27,158,218,191)(28,159,219,192)(29,160,220,187)(30,161,221,188)(31,124,64,91)(32,125,65,92)(33,126,66,93)(34,121,61,94)(35,122,62,95)(36,123,63,96)(37,130,70,97)(38,131,71,98)(39,132,72,99)(40,127,67,100)(41,128,68,101)(42,129,69,102)(43,136,76,103)(44,137,77,104)(45,138,78,105)(46,133,73,106)(47,134,74,107)(48,135,75,108)(49,142,82,109)(50,143,83,110)(51,144,84,111)(52,139,79,112)(53,140,80,113)(54,141,81,114)(145,205,178,238)(146,206,179,239)(147,207,180,240)(148,208,175,235)(149,209,176,236)(150,210,177,237), (1,39,51,48,31)(2,40,52,43,32)(3,41,53,44,33)(4,42,54,45,34)(5,37,49,46,35)(6,38,50,47,36)(7,16,30,237,22)(8,17,25,238,23)(9,18,26,239,24)(10,13,27,240,19)(11,14,28,235,20)(12,15,29,236,21)(55,72,84,75,64)(56,67,79,76,65)(57,68,80,77,66)(58,69,81,78,61)(59,70,82,73,62)(60,71,83,74,63)(85,102,114,105,94)(86,97,109,106,95)(87,98,110,107,96)(88,99,111,108,91)(89,100,112,103,92)(90,101,113,104,93)(115,132,144,135,124)(116,127,139,136,125)(117,128,140,137,126)(118,129,141,138,121)(119,130,142,133,122)(120,131,143,134,123)(145,154,165,174,162)(146,155,166,169,157)(147,156,167,170,158)(148,151,168,171,159)(149,152,163,172,160)(150,153,164,173,161)(175,184,195,204,192)(176,185,196,199,187)(177,186,197,200,188)(178,181,198,201,189)(179,182,193,202,190)(180,183,194,203,191)(205,214,225,234,222)(206,215,226,229,217)(207,216,227,230,218)(208,211,228,231,219)(209,212,223,232,220)(210,213,224,233,221), (1,162,4,159)(2,157,5,160)(3,158,6,161)(7,140,10,143)(8,141,11,144)(9,142,12,139)(13,131,16,128)(14,132,17,129)(15,127,18,130)(19,134,22,137)(20,135,23,138)(21,136,24,133)(25,118,28,115)(26,119,29,116)(27,120,30,117)(31,145,34,148)(32,146,35,149)(33,147,36,150)(37,172,40,169)(38,173,41,170)(39,174,42,171)(43,155,46,152)(44,156,47,153)(45,151,48,154)(49,163,52,166)(50,164,53,167)(51,165,54,168)(55,189,58,192)(56,190,59,187)(57,191,60,188)(61,175,64,178)(62,176,65,179)(63,177,66,180)(67,202,70,199)(68,203,71,200)(69,204,72,201)(73,185,76,182)(74,186,77,183)(75,181,78,184)(79,193,82,196)(80,194,83,197)(81,195,84,198)(85,219,88,222)(86,220,89,217)(87,221,90,218)(91,205,94,208)(92,206,95,209)(93,207,96,210)(97,232,100,229)(98,233,101,230)(99,234,102,231)(103,215,106,212)(104,216,107,213)(105,211,108,214)(109,223,112,226)(110,224,113,227)(111,225,114,228)(121,235,124,238)(122,236,125,239)(123,237,126,240)>;
G:=Group( (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,11)(8,10)(13,17)(14,16)(19,23)(20,22)(25,27)(28,30)(31,63)(32,62)(33,61)(34,66)(35,65)(36,64)(37,67)(38,72)(39,71)(40,70)(41,69)(42,68)(43,73)(44,78)(45,77)(46,76)(47,75)(48,74)(49,79)(50,84)(51,83)(52,82)(53,81)(54,80)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,130)(98,129)(99,128)(100,127)(101,132)(102,131)(103,136)(104,135)(105,134)(106,133)(107,138)(108,137)(109,142)(110,141)(111,140)(112,139)(113,144)(114,143)(145,150)(146,149)(147,148)(151,156)(152,155)(153,154)(157,160)(158,159)(161,162)(163,166)(164,165)(167,168)(169,172)(170,171)(173,174)(175,180)(176,179)(177,178)(181,186)(182,185)(183,184)(187,190)(188,189)(191,192)(193,196)(194,195)(197,198)(199,202)(200,201)(203,204)(205,207)(208,210)(211,213)(214,216)(218,222)(219,221)(224,228)(225,227)(230,234)(231,233)(235,237)(238,240), (1,115,55,88)(2,116,56,89)(3,117,57,90)(4,118,58,85)(5,119,59,86)(6,120,60,87)(7,164,224,197)(8,165,225,198)(9,166,226,193)(10,167,227,194)(11,168,228,195)(12,163,223,196)(13,170,230,203)(14,171,231,204)(15,172,232,199)(16,173,233,200)(17,174,234,201)(18,169,229,202)(19,156,216,183)(20,151,211,184)(21,152,212,185)(22,153,213,186)(23,154,214,181)(24,155,215,182)(25,162,222,189)(26,157,217,190)(27,158,218,191)(28,159,219,192)(29,160,220,187)(30,161,221,188)(31,124,64,91)(32,125,65,92)(33,126,66,93)(34,121,61,94)(35,122,62,95)(36,123,63,96)(37,130,70,97)(38,131,71,98)(39,132,72,99)(40,127,67,100)(41,128,68,101)(42,129,69,102)(43,136,76,103)(44,137,77,104)(45,138,78,105)(46,133,73,106)(47,134,74,107)(48,135,75,108)(49,142,82,109)(50,143,83,110)(51,144,84,111)(52,139,79,112)(53,140,80,113)(54,141,81,114)(145,205,178,238)(146,206,179,239)(147,207,180,240)(148,208,175,235)(149,209,176,236)(150,210,177,237), (1,39,51,48,31)(2,40,52,43,32)(3,41,53,44,33)(4,42,54,45,34)(5,37,49,46,35)(6,38,50,47,36)(7,16,30,237,22)(8,17,25,238,23)(9,18,26,239,24)(10,13,27,240,19)(11,14,28,235,20)(12,15,29,236,21)(55,72,84,75,64)(56,67,79,76,65)(57,68,80,77,66)(58,69,81,78,61)(59,70,82,73,62)(60,71,83,74,63)(85,102,114,105,94)(86,97,109,106,95)(87,98,110,107,96)(88,99,111,108,91)(89,100,112,103,92)(90,101,113,104,93)(115,132,144,135,124)(116,127,139,136,125)(117,128,140,137,126)(118,129,141,138,121)(119,130,142,133,122)(120,131,143,134,123)(145,154,165,174,162)(146,155,166,169,157)(147,156,167,170,158)(148,151,168,171,159)(149,152,163,172,160)(150,153,164,173,161)(175,184,195,204,192)(176,185,196,199,187)(177,186,197,200,188)(178,181,198,201,189)(179,182,193,202,190)(180,183,194,203,191)(205,214,225,234,222)(206,215,226,229,217)(207,216,227,230,218)(208,211,228,231,219)(209,212,223,232,220)(210,213,224,233,221), (1,162,4,159)(2,157,5,160)(3,158,6,161)(7,140,10,143)(8,141,11,144)(9,142,12,139)(13,131,16,128)(14,132,17,129)(15,127,18,130)(19,134,22,137)(20,135,23,138)(21,136,24,133)(25,118,28,115)(26,119,29,116)(27,120,30,117)(31,145,34,148)(32,146,35,149)(33,147,36,150)(37,172,40,169)(38,173,41,170)(39,174,42,171)(43,155,46,152)(44,156,47,153)(45,151,48,154)(49,163,52,166)(50,164,53,167)(51,165,54,168)(55,189,58,192)(56,190,59,187)(57,191,60,188)(61,175,64,178)(62,176,65,179)(63,177,66,180)(67,202,70,199)(68,203,71,200)(69,204,72,201)(73,185,76,182)(74,186,77,183)(75,181,78,184)(79,193,82,196)(80,194,83,197)(81,195,84,198)(85,219,88,222)(86,220,89,217)(87,221,90,218)(91,205,94,208)(92,206,95,209)(93,207,96,210)(97,232,100,229)(98,233,101,230)(99,234,102,231)(103,215,106,212)(104,216,107,213)(105,211,108,214)(109,223,112,226)(110,224,113,227)(111,225,114,228)(121,235,124,238)(122,236,125,239)(123,237,126,240) );
G=PermutationGroup([[(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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,11),(8,10),(13,17),(14,16),(19,23),(20,22),(25,27),(28,30),(31,63),(32,62),(33,61),(34,66),(35,65),(36,64),(37,67),(38,72),(39,71),(40,70),(41,69),(42,68),(43,73),(44,78),(45,77),(46,76),(47,75),(48,74),(49,79),(50,84),(51,83),(52,82),(53,81),(54,80),(85,120),(86,119),(87,118),(88,117),(89,116),(90,115),(91,126),(92,125),(93,124),(94,123),(95,122),(96,121),(97,130),(98,129),(99,128),(100,127),(101,132),(102,131),(103,136),(104,135),(105,134),(106,133),(107,138),(108,137),(109,142),(110,141),(111,140),(112,139),(113,144),(114,143),(145,150),(146,149),(147,148),(151,156),(152,155),(153,154),(157,160),(158,159),(161,162),(163,166),(164,165),(167,168),(169,172),(170,171),(173,174),(175,180),(176,179),(177,178),(181,186),(182,185),(183,184),(187,190),(188,189),(191,192),(193,196),(194,195),(197,198),(199,202),(200,201),(203,204),(205,207),(208,210),(211,213),(214,216),(218,222),(219,221),(224,228),(225,227),(230,234),(231,233),(235,237),(238,240)], [(1,115,55,88),(2,116,56,89),(3,117,57,90),(4,118,58,85),(5,119,59,86),(6,120,60,87),(7,164,224,197),(8,165,225,198),(9,166,226,193),(10,167,227,194),(11,168,228,195),(12,163,223,196),(13,170,230,203),(14,171,231,204),(15,172,232,199),(16,173,233,200),(17,174,234,201),(18,169,229,202),(19,156,216,183),(20,151,211,184),(21,152,212,185),(22,153,213,186),(23,154,214,181),(24,155,215,182),(25,162,222,189),(26,157,217,190),(27,158,218,191),(28,159,219,192),(29,160,220,187),(30,161,221,188),(31,124,64,91),(32,125,65,92),(33,126,66,93),(34,121,61,94),(35,122,62,95),(36,123,63,96),(37,130,70,97),(38,131,71,98),(39,132,72,99),(40,127,67,100),(41,128,68,101),(42,129,69,102),(43,136,76,103),(44,137,77,104),(45,138,78,105),(46,133,73,106),(47,134,74,107),(48,135,75,108),(49,142,82,109),(50,143,83,110),(51,144,84,111),(52,139,79,112),(53,140,80,113),(54,141,81,114),(145,205,178,238),(146,206,179,239),(147,207,180,240),(148,208,175,235),(149,209,176,236),(150,210,177,237)], [(1,39,51,48,31),(2,40,52,43,32),(3,41,53,44,33),(4,42,54,45,34),(5,37,49,46,35),(6,38,50,47,36),(7,16,30,237,22),(8,17,25,238,23),(9,18,26,239,24),(10,13,27,240,19),(11,14,28,235,20),(12,15,29,236,21),(55,72,84,75,64),(56,67,79,76,65),(57,68,80,77,66),(58,69,81,78,61),(59,70,82,73,62),(60,71,83,74,63),(85,102,114,105,94),(86,97,109,106,95),(87,98,110,107,96),(88,99,111,108,91),(89,100,112,103,92),(90,101,113,104,93),(115,132,144,135,124),(116,127,139,136,125),(117,128,140,137,126),(118,129,141,138,121),(119,130,142,133,122),(120,131,143,134,123),(145,154,165,174,162),(146,155,166,169,157),(147,156,167,170,158),(148,151,168,171,159),(149,152,163,172,160),(150,153,164,173,161),(175,184,195,204,192),(176,185,196,199,187),(177,186,197,200,188),(178,181,198,201,189),(179,182,193,202,190),(180,183,194,203,191),(205,214,225,234,222),(206,215,226,229,217),(207,216,227,230,218),(208,211,228,231,219),(209,212,223,232,220),(210,213,224,233,221)], [(1,162,4,159),(2,157,5,160),(3,158,6,161),(7,140,10,143),(8,141,11,144),(9,142,12,139),(13,131,16,128),(14,132,17,129),(15,127,18,130),(19,134,22,137),(20,135,23,138),(21,136,24,133),(25,118,28,115),(26,119,29,116),(27,120,30,117),(31,145,34,148),(32,146,35,149),(33,147,36,150),(37,172,40,169),(38,173,41,170),(39,174,42,171),(43,155,46,152),(44,156,47,153),(45,151,48,154),(49,163,52,166),(50,164,53,167),(51,165,54,168),(55,189,58,192),(56,190,59,187),(57,191,60,188),(61,175,64,178),(62,176,65,179),(63,177,66,180),(67,202,70,199),(68,203,71,200),(69,204,72,201),(73,185,76,182),(74,186,77,183),(75,181,78,184),(79,193,82,196),(80,194,83,197),(81,195,84,198),(85,219,88,222),(86,220,89,217),(87,221,90,218),(91,205,94,208),(92,206,95,209),(93,207,96,210),(97,232,100,229),(98,233,101,230),(99,234,102,231),(103,215,106,212),(104,216,107,213),(105,211,108,214),(109,223,112,226),(110,224,113,227),(111,225,114,228),(121,235,124,238),(122,236,125,239),(123,237,126,240)]])
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 | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 4 | 6 | 6 | 10 | 10 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 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 | 2 | 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 | D10 | C4○D12 | C4○D20 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | C2×S3×D5 | D20⋊5S3 | D12⋊D5 | Dic3.D10 |
| kernel | D6⋊C4.D5 | Dic3×Dic5 | D6⋊Dic5 | C30.Q8 | C3×C4⋊Dic5 | C5×D6⋊C4 | C30.4Q8 | C4⋊Dic5 | D6⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C22×S3 | C10 | C6 | C10 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 6 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D6⋊C4.D5 ►in GL8(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 54 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 50 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 |
G:=sub<GL(8,GF(61))| [0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,21,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,50,48,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,3,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,42,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,60,0,0,0,0,0,0,0,0,36,2,0,0,0,0,0,0,54,25,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,28,11] >;
D6⋊C4.D5 in GAP, Magma, Sage, TeX
D_6\rtimes C_4.D_5
% in TeX
G:=Group("D6:C4.D5"); // GroupNames label
G:=SmallGroup(480,417);
// by ID
G=gap.SmallGroup(480,417);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^4=d^5=1,e^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^3*b,b*d=d*b,e*b*e^-1=b*c^2,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations