Copied to
clipboard

G = (D5×Dic3)⋊C4order 480 = 25·3·5

2nd semidirect product of D5×Dic3 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (D5×Dic3)⋊2C4, (C4×Dic3)⋊13D5, D10.18(C4×S3), (C2×C20).262D6, C33(C42⋊D5), (Dic3×C20)⋊23C2, C6.68(C4○D20), (C2×C12).187D10, (C2×C30).83C23, C30.45(C22×C4), Dic155C414C2, C30.4Q830C2, (C2×Dic5).97D6, Dic3.15(C4×D5), (C22×D5).41D6, C1512(C42⋊C2), (Dic3×Dic5)⋊11C2, D10⋊C4.10S3, C30.109(C4○D4), C2.3(D205S3), (C2×C60).406C22, Dic15.27(C2×C4), D10⋊Dic3.9C2, C52(C23.16D6), C10.25(D42S3), (C2×Dic3).176D10, C2.2(Dic5.D6), (C6×Dic5).48C22, (C2×Dic15).70C22, (C10×Dic3).174C22, C6.13(C2×C4×D5), C2.16(C4×S3×D5), C10.45(S3×C2×C4), (C6×D5).7(C2×C4), (C2×C4).74(S3×D5), (C2×D5×Dic3).2C2, C22.38(C2×S3×D5), (D5×C2×C6).10C22, (C2×C6).95(C22×D5), (C2×C10).95(C22×S3), (C5×Dic3).34(C2×C4), (C3×D10⋊C4).15C2, SmallGroup(480,469)

Series: Derived Chief Lower central Upper central

C1C30 — (D5×Dic3)⋊C4
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — (D5×Dic3)⋊C4
C15C30 — (D5×Dic3)⋊C4
C1C22C2×C4

Generators and relations for (D5×Dic3)⋊C4
 G = < a,b,c,d,e | a5=b2=c6=e4=1, d2=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc3, dcd-1=c-1, ce=ec, de=ed >

Subgroups: 652 in 152 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×8], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×4], C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×Dic3 [×2], C2×Dic3 [×6], C2×C12, C2×C12, C22×C6, C3×D5 [×2], C30 [×3], C42⋊C2, C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4×Dic3, C4×Dic3, Dic3⋊C4 [×2], C6.D4, C3×C22⋊C4, C22×Dic3, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15 [×2], Dic15, C60, C6×D5 [×2], C6×D5 [×2], C2×C30, C4×Dic5, C10.D4 [×2], D10⋊C4, D10⋊C4, C4×C20, C2×C4×D5, C23.16D6, D5×Dic3 [×4], C6×Dic5, C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, C42⋊D5, Dic3×Dic5, D10⋊Dic3, Dic155C4, C3×D10⋊C4, Dic3×C20, C30.4Q8, C2×D5×Dic3, (D5×Dic3)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C4×D5 [×2], C22×D5, S3×C2×C4, D42S3 [×2], S3×D5, C2×C4×D5, C4○D20 [×2], C23.16D6, C2×S3×D5, C42⋊D5, D205S3, C4×S3×D5, Dic5.D6, (D5×Dic3)⋊C4

Smallest permutation representation of (D5×Dic3)⋊C4
On 240 points
Generators in S240
(1 43 53 39 36)(2 44 54 40 31)(3 45 49 41 32)(4 46 50 42 33)(5 47 51 37 34)(6 48 52 38 35)(7 26 19 237 17)(8 27 20 238 18)(9 28 21 239 13)(10 29 22 240 14)(11 30 23 235 15)(12 25 24 236 16)(55 78 82 68 65)(56 73 83 69 66)(57 74 84 70 61)(58 75 79 71 62)(59 76 80 72 63)(60 77 81 67 64)(85 108 112 98 95)(86 103 113 99 96)(87 104 114 100 91)(88 105 109 101 92)(89 106 110 102 93)(90 107 111 97 94)(115 138 142 123 130)(116 133 143 124 131)(117 134 144 125 132)(118 135 139 126 127)(119 136 140 121 128)(120 137 141 122 129)(145 164 170 153 162)(146 165 171 154 157)(147 166 172 155 158)(148 167 173 156 159)(149 168 174 151 160)(150 163 169 152 161)(175 194 200 183 192)(176 195 201 184 187)(177 196 202 185 188)(178 197 203 186 189)(179 198 204 181 190)(180 193 199 182 191)(205 224 230 213 222)(206 225 231 214 217)(207 226 232 215 218)(208 227 233 216 219)(209 228 234 211 220)(210 223 229 212 221)
(1 66)(2 61)(3 62)(4 63)(5 64)(6 65)(7 229)(8 230)(9 231)(10 232)(11 233)(12 234)(13 214)(14 215)(15 216)(16 211)(17 212)(18 213)(19 210)(20 205)(21 206)(22 207)(23 208)(24 209)(25 228)(26 223)(27 224)(28 225)(29 226)(30 227)(31 57)(32 58)(33 59)(34 60)(35 55)(36 56)(37 77)(38 78)(39 73)(40 74)(41 75)(42 76)(43 69)(44 70)(45 71)(46 72)(47 67)(48 68)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 120)(92 115)(93 116)(94 117)(95 118)(96 119)(97 134)(98 135)(99 136)(100 137)(101 138)(102 133)(103 121)(104 122)(105 123)(106 124)(107 125)(108 126)(109 142)(110 143)(111 144)(112 139)(113 140)(114 141)(145 192)(146 187)(147 188)(148 189)(149 190)(150 191)(151 198)(152 193)(153 194)(154 195)(155 196)(156 197)(157 176)(158 177)(159 178)(160 179)(161 180)(162 175)(163 182)(164 183)(165 184)(166 185)(167 186)(168 181)(169 199)(170 200)(171 201)(172 202)(173 203)(174 204)(217 239)(218 240)(219 235)(220 236)(221 237)(222 238)
(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)
(1 145 4 148)(2 150 5 147)(3 149 6 146)(7 144 10 141)(8 143 11 140)(9 142 12 139)(13 138 16 135)(14 137 17 134)(15 136 18 133)(19 132 22 129)(20 131 23 128)(21 130 24 127)(25 126 28 123)(26 125 29 122)(27 124 30 121)(31 161 34 158)(32 160 35 157)(33 159 36 162)(37 155 40 152)(38 154 41 151)(39 153 42 156)(43 164 46 167)(44 163 47 166)(45 168 48 165)(49 174 52 171)(50 173 53 170)(51 172 54 169)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 191 64 188)(62 190 65 187)(63 189 66 192)(67 185 70 182)(68 184 71 181)(69 183 72 186)(73 194 76 197)(74 193 77 196)(75 198 78 195)(79 204 82 201)(80 203 83 200)(81 202 84 199)(85 206 88 209)(86 205 89 208)(87 210 90 207)(91 221 94 218)(92 220 95 217)(93 219 96 222)(97 215 100 212)(98 214 101 211)(99 213 102 216)(103 224 106 227)(104 223 107 226)(105 228 108 225)(109 234 112 231)(110 233 113 230)(111 232 114 229)(115 236 118 239)(116 235 119 238)(117 240 120 237)
(1 116 56 86)(2 117 57 87)(3 118 58 88)(4 119 59 89)(5 120 60 90)(6 115 55 85)(7 202 232 172)(8 203 233 173)(9 204 234 174)(10 199 229 169)(11 200 230 170)(12 201 231 171)(13 198 228 168)(14 193 223 163)(15 194 224 164)(16 195 225 165)(17 196 226 166)(18 197 227 167)(19 188 218 158)(20 189 219 159)(21 190 220 160)(22 191 221 161)(23 192 222 162)(24 187 217 157)(25 184 214 154)(26 185 215 155)(27 186 216 156)(28 181 211 151)(29 182 212 152)(30 183 213 153)(31 132 61 91)(32 127 62 92)(33 128 63 93)(34 129 64 94)(35 130 65 95)(36 131 66 96)(37 122 67 97)(38 123 68 98)(39 124 69 99)(40 125 70 100)(41 126 71 101)(42 121 72 102)(43 133 73 103)(44 134 74 104)(45 135 75 105)(46 136 76 106)(47 137 77 107)(48 138 78 108)(49 139 79 109)(50 140 80 110)(51 141 81 111)(52 142 82 112)(53 143 83 113)(54 144 84 114)(145 235 175 205)(146 236 176 206)(147 237 177 207)(148 238 178 208)(149 239 179 209)(150 240 180 210)

G:=sub<Sym(240)| (1,43,53,39,36)(2,44,54,40,31)(3,45,49,41,32)(4,46,50,42,33)(5,47,51,37,34)(6,48,52,38,35)(7,26,19,237,17)(8,27,20,238,18)(9,28,21,239,13)(10,29,22,240,14)(11,30,23,235,15)(12,25,24,236,16)(55,78,82,68,65)(56,73,83,69,66)(57,74,84,70,61)(58,75,79,71,62)(59,76,80,72,63)(60,77,81,67,64)(85,108,112,98,95)(86,103,113,99,96)(87,104,114,100,91)(88,105,109,101,92)(89,106,110,102,93)(90,107,111,97,94)(115,138,142,123,130)(116,133,143,124,131)(117,134,144,125,132)(118,135,139,126,127)(119,136,140,121,128)(120,137,141,122,129)(145,164,170,153,162)(146,165,171,154,157)(147,166,172,155,158)(148,167,173,156,159)(149,168,174,151,160)(150,163,169,152,161)(175,194,200,183,192)(176,195,201,184,187)(177,196,202,185,188)(178,197,203,186,189)(179,198,204,181,190)(180,193,199,182,191)(205,224,230,213,222)(206,225,231,214,217)(207,226,232,215,218)(208,227,233,216,219)(209,228,234,211,220)(210,223,229,212,221), (1,66)(2,61)(3,62)(4,63)(5,64)(6,65)(7,229)(8,230)(9,231)(10,232)(11,233)(12,234)(13,214)(14,215)(15,216)(16,211)(17,212)(18,213)(19,210)(20,205)(21,206)(22,207)(23,208)(24,209)(25,228)(26,223)(27,224)(28,225)(29,226)(30,227)(31,57)(32,58)(33,59)(34,60)(35,55)(36,56)(37,77)(38,78)(39,73)(40,74)(41,75)(42,76)(43,69)(44,70)(45,71)(46,72)(47,67)(48,68)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,120)(92,115)(93,116)(94,117)(95,118)(96,119)(97,134)(98,135)(99,136)(100,137)(101,138)(102,133)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,142)(110,143)(111,144)(112,139)(113,140)(114,141)(145,192)(146,187)(147,188)(148,189)(149,190)(150,191)(151,198)(152,193)(153,194)(154,195)(155,196)(156,197)(157,176)(158,177)(159,178)(160,179)(161,180)(162,175)(163,182)(164,183)(165,184)(166,185)(167,186)(168,181)(169,199)(170,200)(171,201)(172,202)(173,203)(174,204)(217,239)(218,240)(219,235)(220,236)(221,237)(222,238), (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), (1,145,4,148)(2,150,5,147)(3,149,6,146)(7,144,10,141)(8,143,11,140)(9,142,12,139)(13,138,16,135)(14,137,17,134)(15,136,18,133)(19,132,22,129)(20,131,23,128)(21,130,24,127)(25,126,28,123)(26,125,29,122)(27,124,30,121)(31,161,34,158)(32,160,35,157)(33,159,36,162)(37,155,40,152)(38,154,41,151)(39,153,42,156)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,174,52,171)(50,173,53,170)(51,172,54,169)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,185,70,182)(68,184,71,181)(69,183,72,186)(73,194,76,197)(74,193,77,196)(75,198,78,195)(79,204,82,201)(80,203,83,200)(81,202,84,199)(85,206,88,209)(86,205,89,208)(87,210,90,207)(91,221,94,218)(92,220,95,217)(93,219,96,222)(97,215,100,212)(98,214,101,211)(99,213,102,216)(103,224,106,227)(104,223,107,226)(105,228,108,225)(109,234,112,231)(110,233,113,230)(111,232,114,229)(115,236,118,239)(116,235,119,238)(117,240,120,237), (1,116,56,86)(2,117,57,87)(3,118,58,88)(4,119,59,89)(5,120,60,90)(6,115,55,85)(7,202,232,172)(8,203,233,173)(9,204,234,174)(10,199,229,169)(11,200,230,170)(12,201,231,171)(13,198,228,168)(14,193,223,163)(15,194,224,164)(16,195,225,165)(17,196,226,166)(18,197,227,167)(19,188,218,158)(20,189,219,159)(21,190,220,160)(22,191,221,161)(23,192,222,162)(24,187,217,157)(25,184,214,154)(26,185,215,155)(27,186,216,156)(28,181,211,151)(29,182,212,152)(30,183,213,153)(31,132,61,91)(32,127,62,92)(33,128,63,93)(34,129,64,94)(35,130,65,95)(36,131,66,96)(37,122,67,97)(38,123,68,98)(39,124,69,99)(40,125,70,100)(41,126,71,101)(42,121,72,102)(43,133,73,103)(44,134,74,104)(45,135,75,105)(46,136,76,106)(47,137,77,107)(48,138,78,108)(49,139,79,109)(50,140,80,110)(51,141,81,111)(52,142,82,112)(53,143,83,113)(54,144,84,114)(145,235,175,205)(146,236,176,206)(147,237,177,207)(148,238,178,208)(149,239,179,209)(150,240,180,210)>;

G:=Group( (1,43,53,39,36)(2,44,54,40,31)(3,45,49,41,32)(4,46,50,42,33)(5,47,51,37,34)(6,48,52,38,35)(7,26,19,237,17)(8,27,20,238,18)(9,28,21,239,13)(10,29,22,240,14)(11,30,23,235,15)(12,25,24,236,16)(55,78,82,68,65)(56,73,83,69,66)(57,74,84,70,61)(58,75,79,71,62)(59,76,80,72,63)(60,77,81,67,64)(85,108,112,98,95)(86,103,113,99,96)(87,104,114,100,91)(88,105,109,101,92)(89,106,110,102,93)(90,107,111,97,94)(115,138,142,123,130)(116,133,143,124,131)(117,134,144,125,132)(118,135,139,126,127)(119,136,140,121,128)(120,137,141,122,129)(145,164,170,153,162)(146,165,171,154,157)(147,166,172,155,158)(148,167,173,156,159)(149,168,174,151,160)(150,163,169,152,161)(175,194,200,183,192)(176,195,201,184,187)(177,196,202,185,188)(178,197,203,186,189)(179,198,204,181,190)(180,193,199,182,191)(205,224,230,213,222)(206,225,231,214,217)(207,226,232,215,218)(208,227,233,216,219)(209,228,234,211,220)(210,223,229,212,221), (1,66)(2,61)(3,62)(4,63)(5,64)(6,65)(7,229)(8,230)(9,231)(10,232)(11,233)(12,234)(13,214)(14,215)(15,216)(16,211)(17,212)(18,213)(19,210)(20,205)(21,206)(22,207)(23,208)(24,209)(25,228)(26,223)(27,224)(28,225)(29,226)(30,227)(31,57)(32,58)(33,59)(34,60)(35,55)(36,56)(37,77)(38,78)(39,73)(40,74)(41,75)(42,76)(43,69)(44,70)(45,71)(46,72)(47,67)(48,68)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,120)(92,115)(93,116)(94,117)(95,118)(96,119)(97,134)(98,135)(99,136)(100,137)(101,138)(102,133)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(109,142)(110,143)(111,144)(112,139)(113,140)(114,141)(145,192)(146,187)(147,188)(148,189)(149,190)(150,191)(151,198)(152,193)(153,194)(154,195)(155,196)(156,197)(157,176)(158,177)(159,178)(160,179)(161,180)(162,175)(163,182)(164,183)(165,184)(166,185)(167,186)(168,181)(169,199)(170,200)(171,201)(172,202)(173,203)(174,204)(217,239)(218,240)(219,235)(220,236)(221,237)(222,238), (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), (1,145,4,148)(2,150,5,147)(3,149,6,146)(7,144,10,141)(8,143,11,140)(9,142,12,139)(13,138,16,135)(14,137,17,134)(15,136,18,133)(19,132,22,129)(20,131,23,128)(21,130,24,127)(25,126,28,123)(26,125,29,122)(27,124,30,121)(31,161,34,158)(32,160,35,157)(33,159,36,162)(37,155,40,152)(38,154,41,151)(39,153,42,156)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,174,52,171)(50,173,53,170)(51,172,54,169)(55,176,58,179)(56,175,59,178)(57,180,60,177)(61,191,64,188)(62,190,65,187)(63,189,66,192)(67,185,70,182)(68,184,71,181)(69,183,72,186)(73,194,76,197)(74,193,77,196)(75,198,78,195)(79,204,82,201)(80,203,83,200)(81,202,84,199)(85,206,88,209)(86,205,89,208)(87,210,90,207)(91,221,94,218)(92,220,95,217)(93,219,96,222)(97,215,100,212)(98,214,101,211)(99,213,102,216)(103,224,106,227)(104,223,107,226)(105,228,108,225)(109,234,112,231)(110,233,113,230)(111,232,114,229)(115,236,118,239)(116,235,119,238)(117,240,120,237), (1,116,56,86)(2,117,57,87)(3,118,58,88)(4,119,59,89)(5,120,60,90)(6,115,55,85)(7,202,232,172)(8,203,233,173)(9,204,234,174)(10,199,229,169)(11,200,230,170)(12,201,231,171)(13,198,228,168)(14,193,223,163)(15,194,224,164)(16,195,225,165)(17,196,226,166)(18,197,227,167)(19,188,218,158)(20,189,219,159)(21,190,220,160)(22,191,221,161)(23,192,222,162)(24,187,217,157)(25,184,214,154)(26,185,215,155)(27,186,216,156)(28,181,211,151)(29,182,212,152)(30,183,213,153)(31,132,61,91)(32,127,62,92)(33,128,63,93)(34,129,64,94)(35,130,65,95)(36,131,66,96)(37,122,67,97)(38,123,68,98)(39,124,69,99)(40,125,70,100)(41,126,71,101)(42,121,72,102)(43,133,73,103)(44,134,74,104)(45,135,75,105)(46,136,76,106)(47,137,77,107)(48,138,78,108)(49,139,79,109)(50,140,80,110)(51,141,81,111)(52,142,82,112)(53,143,83,113)(54,144,84,114)(145,235,175,205)(146,236,176,206)(147,237,177,207)(148,238,178,208)(149,239,179,209)(150,240,180,210) );

G=PermutationGroup([(1,43,53,39,36),(2,44,54,40,31),(3,45,49,41,32),(4,46,50,42,33),(5,47,51,37,34),(6,48,52,38,35),(7,26,19,237,17),(8,27,20,238,18),(9,28,21,239,13),(10,29,22,240,14),(11,30,23,235,15),(12,25,24,236,16),(55,78,82,68,65),(56,73,83,69,66),(57,74,84,70,61),(58,75,79,71,62),(59,76,80,72,63),(60,77,81,67,64),(85,108,112,98,95),(86,103,113,99,96),(87,104,114,100,91),(88,105,109,101,92),(89,106,110,102,93),(90,107,111,97,94),(115,138,142,123,130),(116,133,143,124,131),(117,134,144,125,132),(118,135,139,126,127),(119,136,140,121,128),(120,137,141,122,129),(145,164,170,153,162),(146,165,171,154,157),(147,166,172,155,158),(148,167,173,156,159),(149,168,174,151,160),(150,163,169,152,161),(175,194,200,183,192),(176,195,201,184,187),(177,196,202,185,188),(178,197,203,186,189),(179,198,204,181,190),(180,193,199,182,191),(205,224,230,213,222),(206,225,231,214,217),(207,226,232,215,218),(208,227,233,216,219),(209,228,234,211,220),(210,223,229,212,221)], [(1,66),(2,61),(3,62),(4,63),(5,64),(6,65),(7,229),(8,230),(9,231),(10,232),(11,233),(12,234),(13,214),(14,215),(15,216),(16,211),(17,212),(18,213),(19,210),(20,205),(21,206),(22,207),(23,208),(24,209),(25,228),(26,223),(27,224),(28,225),(29,226),(30,227),(31,57),(32,58),(33,59),(34,60),(35,55),(36,56),(37,77),(38,78),(39,73),(40,74),(41,75),(42,76),(43,69),(44,70),(45,71),(46,72),(47,67),(48,68),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(85,127),(86,128),(87,129),(88,130),(89,131),(90,132),(91,120),(92,115),(93,116),(94,117),(95,118),(96,119),(97,134),(98,135),(99,136),(100,137),(101,138),(102,133),(103,121),(104,122),(105,123),(106,124),(107,125),(108,126),(109,142),(110,143),(111,144),(112,139),(113,140),(114,141),(145,192),(146,187),(147,188),(148,189),(149,190),(150,191),(151,198),(152,193),(153,194),(154,195),(155,196),(156,197),(157,176),(158,177),(159,178),(160,179),(161,180),(162,175),(163,182),(164,183),(165,184),(166,185),(167,186),(168,181),(169,199),(170,200),(171,201),(172,202),(173,203),(174,204),(217,239),(218,240),(219,235),(220,236),(221,237),(222,238)], [(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)], [(1,145,4,148),(2,150,5,147),(3,149,6,146),(7,144,10,141),(8,143,11,140),(9,142,12,139),(13,138,16,135),(14,137,17,134),(15,136,18,133),(19,132,22,129),(20,131,23,128),(21,130,24,127),(25,126,28,123),(26,125,29,122),(27,124,30,121),(31,161,34,158),(32,160,35,157),(33,159,36,162),(37,155,40,152),(38,154,41,151),(39,153,42,156),(43,164,46,167),(44,163,47,166),(45,168,48,165),(49,174,52,171),(50,173,53,170),(51,172,54,169),(55,176,58,179),(56,175,59,178),(57,180,60,177),(61,191,64,188),(62,190,65,187),(63,189,66,192),(67,185,70,182),(68,184,71,181),(69,183,72,186),(73,194,76,197),(74,193,77,196),(75,198,78,195),(79,204,82,201),(80,203,83,200),(81,202,84,199),(85,206,88,209),(86,205,89,208),(87,210,90,207),(91,221,94,218),(92,220,95,217),(93,219,96,222),(97,215,100,212),(98,214,101,211),(99,213,102,216),(103,224,106,227),(104,223,107,226),(105,228,108,225),(109,234,112,231),(110,233,113,230),(111,232,114,229),(115,236,118,239),(116,235,119,238),(117,240,120,237)], [(1,116,56,86),(2,117,57,87),(3,118,58,88),(4,119,59,89),(5,120,60,90),(6,115,55,85),(7,202,232,172),(8,203,233,173),(9,204,234,174),(10,199,229,169),(11,200,230,170),(12,201,231,171),(13,198,228,168),(14,193,223,163),(15,194,224,164),(16,195,225,165),(17,196,226,166),(18,197,227,167),(19,188,218,158),(20,189,219,159),(21,190,220,160),(22,191,221,161),(23,192,222,162),(24,187,217,157),(25,184,214,154),(26,185,215,155),(27,186,216,156),(28,181,211,151),(29,182,212,152),(30,183,213,153),(31,132,61,91),(32,127,62,92),(33,128,63,93),(34,129,64,94),(35,130,65,95),(36,131,66,96),(37,122,67,97),(38,123,68,98),(39,124,69,99),(40,125,70,100),(41,126,71,101),(42,121,72,102),(43,133,73,103),(44,134,74,104),(45,135,75,105),(46,136,76,106),(47,137,77,107),(48,138,78,108),(49,139,79,109),(50,140,80,110),(51,141,81,111),(52,142,82,112),(53,143,83,113),(54,144,84,114),(145,235,175,205),(146,236,176,206),(147,237,177,207),(148,238,178,208),(149,239,179,209),(150,240,180,210)])

78 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A···20H20I···20X30A···30F60A···60H
order122222344444444444444556666610···1012121212151520···2020···2030···3060···60
size111110102223333661010303030302222220202···2442020442···26···64···44···4

78 irreducible representations

dim11111111122222222222444444
type+++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C4S3D5D6D6D6C4○D4D10D10C4×S3C4×D5C4○D20D42S3S3×D5C2×S3×D5D205S3C4×S3×D5Dic5.D6
kernel(D5×Dic3)⋊C4Dic3×Dic5D10⋊Dic3Dic155C4C3×D10⋊C4Dic3×C20C30.4Q8C2×D5×Dic3D5×Dic3D10⋊C4C4×Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12D10Dic3C6C10C2×C4C22C2C2C2
# reps111111118121114424816222444

Matrix representation of (D5×Dic3)⋊C4 in GL6(𝔽61)

100000
010000
00436000
001000
000010
000001
,
100000
7600000
0018100
00434300
000010
000001
,
6000000
0600000
001000
000100
00005915
0000121
,
1100000
0110000
001000
000100
00003536
00002726
,
1350000
3480000
0011000
0001100
0000600
0000060

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,7,0,0,0,0,0,60,0,0,0,0,0,0,18,43,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,12,0,0,0,0,15,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,27,0,0,0,0,36,26],[13,3,0,0,0,0,5,48,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

(D5×Dic3)⋊C4 in GAP, Magma, Sage, TeX

(D_5\times {\rm Dic}_3)\rtimes C_4
% in TeX

G:=Group("(D5xDic3):C4");
// GroupNames label

G:=SmallGroup(480,469);
// by ID

G=gap.SmallGroup(480,469);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^6=e^4=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^3,d*c*d^-1=c^-1,c*e=e*c,d*e=e*d>;
// generators/relations

׿
×
𝔽