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## G = Dic3×C5⋊D4order 480 = 25·3·5

### Direct product of Dic3 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic3×C5⋊D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×D5×Dic3 — Dic3×C5⋊D4
 Lower central C15 — C30 — Dic3×C5⋊D4
 Upper central C1 — C22 — C23

Generators and relations for Dic3×C5⋊D4
G = < a,b,c,d,e | a6=c5=d4=e2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 764 in 188 conjugacy classes, 72 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×9], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], C20 [×3], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12, C3×D4 [×4], C22×C6, C22×C6, C3×D5 [×2], C30 [×3], C30 [×2], C4×D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×4], C22×D5, C22×C10, C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3, C22×Dic3, C6×D4, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, D4×Dic3, D5×Dic3 [×2], C6×Dic5, C3×C5⋊D4 [×4], C10×Dic3 [×2], C10×Dic3 [×2], C2×Dic15 [×2], D5×C2×C6, C22×C30, C4×C5⋊D4, Dic3×Dic5, D10⋊Dic3, C30.Q8, C30.38D4, C2×D5×Dic3, C6×C5⋊D4, Dic3×C2×C10, Dic3×C5⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C2×Dic3 [×6], C22×S3, C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, S3×D4, D42S3, C22×Dic3, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, D4×Dic3, D5×Dic3 [×2], C2×S3×D5, C4×C5⋊D4, C2×D5×Dic3, Dic5.D6, S3×C5⋊D4, Dic3×C5⋊D4

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

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10N 12A 12B 15A 15B 20A ··· 20P 30A ··· 30N order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 12 12 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 2 2 10 10 2 3 3 3 3 6 6 10 10 30 30 30 30 2 2 2 2 2 4 4 20 20 2 ··· 2 20 20 4 4 6 ··· 6 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 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + - + + + + + - + - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 Dic3 D6 D6 C4○D4 D10 D10 C5⋊D4 C4×D5 C4○D20 S3×D4 D4⋊2S3 S3×D5 D5×Dic3 C2×S3×D5 Dic5.D6 S3×C5⋊D4 kernel Dic3×C5⋊D4 Dic3×Dic5 D10⋊Dic3 C30.Q8 C30.38D4 C2×D5×Dic3 C6×C5⋊D4 Dic3×C2×C10 C3×C5⋊D4 C2×C5⋊D4 C5×Dic3 C22×Dic3 C2×Dic5 C5⋊D4 C22×D5 C22×C10 C30 C2×Dic3 C22×C6 Dic3 C2×C6 C6 C10 C10 C23 C22 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 4 1 1 2 4 2 8 8 8 1 1 2 4 2 4 4

Matrix representation of Dic3×C5⋊D4 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 27 24 0 0 0 0 51 34
,
 17 60 0 0 0 0 1 0 0 0 0 0 0 0 60 60 0 0 0 0 45 44 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 47 16 0 0 0 0 22 14 0 0 0 0 0 0 44 43 0 0 0 0 16 17 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 17 60 0 0 0 0 0 0 44 43 0 0 0 0 16 17 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,27,51,0,0,0,0,24,34],[17,1,0,0,0,0,60,0,0,0,0,0,0,0,60,45,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[47,22,0,0,0,0,16,14,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3×C5⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_5\rtimes D_4
% in TeX

G:=Group("Dic3xC5:D4");
// GroupNames label

G:=SmallGroup(480,629);
// by ID

G=gap.SmallGroup(480,629);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,422,1356,18822]);
// Polycyclic

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

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