Copied to
clipboard

## G = (D5×Dic3)⋊C4order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — (D5×Dic3)⋊C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — (D5×Dic3)⋊C4
 Lower central C15 — C30 — (D5×Dic3)⋊C4
 Upper central C1 — C22 — C2×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, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4×Dic3, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C6×D5, C6×D5, C2×C30, C4×Dic5, C10.D4, D10⋊C4, D10⋊C4, C4×C20, C2×C4×D5, C23.16D6, D5×Dic3, C6×Dic5, C10×Dic3, C2×Dic15, 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, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, D10, C4×S3, C22×S3, C42⋊C2, C4×D5, C22×D5, S3×C2×C4, D42S3, S3×D5, C2×C4×D5, C4○D20, 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 38 53 44 36)(2 39 54 45 31)(3 40 49 46 32)(4 41 50 47 33)(5 42 51 48 34)(6 37 52 43 35)(7 16 19 237 27)(8 17 20 238 28)(9 18 21 239 29)(10 13 22 240 30)(11 14 23 235 25)(12 15 24 236 26)(55 67 82 73 65)(56 68 83 74 66)(57 69 84 75 61)(58 70 79 76 62)(59 71 80 77 63)(60 72 81 78 64)(85 97 112 103 95)(86 98 113 104 96)(87 99 114 105 91)(88 100 109 106 92)(89 101 110 107 93)(90 102 111 108 94)(115 122 142 133 130)(116 123 143 134 131)(117 124 144 135 132)(118 125 139 136 127)(119 126 140 137 128)(120 121 141 138 129)(145 154 170 163 162)(146 155 171 164 157)(147 156 172 165 158)(148 151 173 166 159)(149 152 174 167 160)(150 153 169 168 161)(175 184 200 193 192)(176 185 201 194 187)(177 186 202 195 188)(178 181 203 196 189)(179 182 204 197 190)(180 183 199 198 191)(205 214 230 223 222)(206 215 231 224 217)(207 216 232 225 218)(208 211 233 226 219)(209 212 234 227 220)(210 213 229 228 221)
(1 66)(2 61)(3 62)(4 63)(5 64)(6 65)(7 229)(8 230)(9 231)(10 232)(11 233)(12 234)(13 216)(14 211)(15 212)(16 213)(17 214)(18 215)(19 210)(20 205)(21 206)(22 207)(23 208)(24 209)(25 226)(26 227)(27 228)(28 223)(29 224)(30 225)(31 57)(32 58)(33 59)(34 60)(35 55)(36 56)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(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)(97 136)(98 137)(99 138)(100 133)(101 134)(102 135)(103 125)(104 126)(105 121)(106 122)(107 123)(108 124)(109 142)(110 143)(111 144)(112 139)(113 140)(114 141)(145 192)(146 187)(147 188)(148 189)(149 190)(150 191)(151 196)(152 197)(153 198)(154 193)(155 194)(156 195)(157 176)(158 177)(159 178)(160 179)(161 180)(162 175)(163 184)(164 185)(165 186)(166 181)(167 182)(168 183)(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)| (1,38,53,44,36)(2,39,54,45,31)(3,40,49,46,32)(4,41,50,47,33)(5,42,51,48,34)(6,37,52,43,35)(7,16,19,237,27)(8,17,20,238,28)(9,18,21,239,29)(10,13,22,240,30)(11,14,23,235,25)(12,15,24,236,26)(55,67,82,73,65)(56,68,83,74,66)(57,69,84,75,61)(58,70,79,76,62)(59,71,80,77,63)(60,72,81,78,64)(85,97,112,103,95)(86,98,113,104,96)(87,99,114,105,91)(88,100,109,106,92)(89,101,110,107,93)(90,102,111,108,94)(115,122,142,133,130)(116,123,143,134,131)(117,124,144,135,132)(118,125,139,136,127)(119,126,140,137,128)(120,121,141,138,129)(145,154,170,163,162)(146,155,171,164,157)(147,156,172,165,158)(148,151,173,166,159)(149,152,174,167,160)(150,153,169,168,161)(175,184,200,193,192)(176,185,201,194,187)(177,186,202,195,188)(178,181,203,196,189)(179,182,204,197,190)(180,183,199,198,191)(205,214,230,223,222)(206,215,231,224,217)(207,216,232,225,218)(208,211,233,226,219)(209,212,234,227,220)(210,213,229,228,221), (1,66)(2,61)(3,62)(4,63)(5,64)(6,65)(7,229)(8,230)(9,231)(10,232)(11,233)(12,234)(13,216)(14,211)(15,212)(16,213)(17,214)(18,215)(19,210)(20,205)(21,206)(22,207)(23,208)(24,209)(25,226)(26,227)(27,228)(28,223)(29,224)(30,225)(31,57)(32,58)(33,59)(34,60)(35,55)(36,56)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(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)(97,136)(98,137)(99,138)(100,133)(101,134)(102,135)(103,125)(104,126)(105,121)(106,122)(107,123)(108,124)(109,142)(110,143)(111,144)(112,139)(113,140)(114,141)(145,192)(146,187)(147,188)(148,189)(149,190)(150,191)(151,196)(152,197)(153,198)(154,193)(155,194)(156,195)(157,176)(158,177)(159,178)(160,179)(161,180)(162,175)(163,184)(164,185)(165,186)(166,181)(167,182)(168,183)(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( (1,38,53,44,36)(2,39,54,45,31)(3,40,49,46,32)(4,41,50,47,33)(5,42,51,48,34)(6,37,52,43,35)(7,16,19,237,27)(8,17,20,238,28)(9,18,21,239,29)(10,13,22,240,30)(11,14,23,235,25)(12,15,24,236,26)(55,67,82,73,65)(56,68,83,74,66)(57,69,84,75,61)(58,70,79,76,62)(59,71,80,77,63)(60,72,81,78,64)(85,97,112,103,95)(86,98,113,104,96)(87,99,114,105,91)(88,100,109,106,92)(89,101,110,107,93)(90,102,111,108,94)(115,122,142,133,130)(116,123,143,134,131)(117,124,144,135,132)(118,125,139,136,127)(119,126,140,137,128)(120,121,141,138,129)(145,154,170,163,162)(146,155,171,164,157)(147,156,172,165,158)(148,151,173,166,159)(149,152,174,167,160)(150,153,169,168,161)(175,184,200,193,192)(176,185,201,194,187)(177,186,202,195,188)(178,181,203,196,189)(179,182,204,197,190)(180,183,199,198,191)(205,214,230,223,222)(206,215,231,224,217)(207,216,232,225,218)(208,211,233,226,219)(209,212,234,227,220)(210,213,229,228,221), (1,66)(2,61)(3,62)(4,63)(5,64)(6,65)(7,229)(8,230)(9,231)(10,232)(11,233)(12,234)(13,216)(14,211)(15,212)(16,213)(17,214)(18,215)(19,210)(20,205)(21,206)(22,207)(23,208)(24,209)(25,226)(26,227)(27,228)(28,223)(29,224)(30,225)(31,57)(32,58)(33,59)(34,60)(35,55)(36,56)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(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)(97,136)(98,137)(99,138)(100,133)(101,134)(102,135)(103,125)(104,126)(105,121)(106,122)(107,123)(108,124)(109,142)(110,143)(111,144)(112,139)(113,140)(114,141)(145,192)(146,187)(147,188)(148,189)(149,190)(150,191)(151,196)(152,197)(153,198)(154,193)(155,194)(156,195)(157,176)(158,177)(159,178)(160,179)(161,180)(162,175)(163,184)(164,185)(165,186)(166,181)(167,182)(168,183)(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([[(1,38,53,44,36),(2,39,54,45,31),(3,40,49,46,32),(4,41,50,47,33),(5,42,51,48,34),(6,37,52,43,35),(7,16,19,237,27),(8,17,20,238,28),(9,18,21,239,29),(10,13,22,240,30),(11,14,23,235,25),(12,15,24,236,26),(55,67,82,73,65),(56,68,83,74,66),(57,69,84,75,61),(58,70,79,76,62),(59,71,80,77,63),(60,72,81,78,64),(85,97,112,103,95),(86,98,113,104,96),(87,99,114,105,91),(88,100,109,106,92),(89,101,110,107,93),(90,102,111,108,94),(115,122,142,133,130),(116,123,143,134,131),(117,124,144,135,132),(118,125,139,136,127),(119,126,140,137,128),(120,121,141,138,129),(145,154,170,163,162),(146,155,171,164,157),(147,156,172,165,158),(148,151,173,166,159),(149,152,174,167,160),(150,153,169,168,161),(175,184,200,193,192),(176,185,201,194,187),(177,186,202,195,188),(178,181,203,196,189),(179,182,204,197,190),(180,183,199,198,191),(205,214,230,223,222),(206,215,231,224,217),(207,216,232,225,218),(208,211,233,226,219),(209,212,234,227,220),(210,213,229,228,221)], [(1,66),(2,61),(3,62),(4,63),(5,64),(6,65),(7,229),(8,230),(9,231),(10,232),(11,233),(12,234),(13,216),(14,211),(15,212),(16,213),(17,214),(18,215),(19,210),(20,205),(21,206),(22,207),(23,208),(24,209),(25,226),(26,227),(27,228),(28,223),(29,224),(30,225),(31,57),(32,58),(33,59),(34,60),(35,55),(36,56),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(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),(97,136),(98,137),(99,138),(100,133),(101,134),(102,135),(103,125),(104,126),(105,121),(106,122),(107,123),(108,124),(109,142),(110,143),(111,144),(112,139),(113,140),(114,141),(145,192),(146,187),(147,188),(148,189),(149,190),(150,191),(151,196),(152,197),(153,198),(154,193),(155,194),(156,195),(157,176),(158,177),(159,178),(160,179),(161,180),(162,175),(163,184),(164,185),(165,186),(166,181),(167,182),(168,183),(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 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 12A 12B 12C 12D 15A 15B 20A ··· 20H 20I ··· 20X 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 2 2 2 3 3 3 3 6 6 10 10 30 30 30 30 2 2 2 2 2 20 20 2 ··· 2 4 4 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D5 D6 D6 D6 C4○D4 D10 D10 C4×S3 C4×D5 C4○D20 D4⋊2S3 S3×D5 C2×S3×D5 D20⋊5S3 C4×S3×D5 Dic5.D6 kernel (D5×Dic3)⋊C4 Dic3×Dic5 D10⋊Dic3 Dic15⋊5C4 C3×D10⋊C4 Dic3×C20 C30.4Q8 C2×D5×Dic3 D5×Dic3 D10⋊C4 C4×Dic3 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 D10 Dic3 C6 C10 C2×C4 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 4 4 2 4 8 16 2 2 2 4 4 4

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 43 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 7 60 0 0 0 0 0 0 18 1 0 0 0 0 43 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 15 0 0 0 0 12 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 36 0 0 0 0 27 26
,
 13 5 0 0 0 0 3 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

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