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G = Dic15.31D4order 480 = 25·3·5

12nd non-split extension by Dic15 of D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic15.31D4, D6⋊C413D5, C6.48(D4×D5), C30.64(C2×D4), C10.49(S3×D4), D6⋊Dic520C2, (C2×C20).204D6, D10⋊C413S3, (C4×Dic15)⋊13C2, C6.42(C4○D20), (C2×C12).202D10, C1512(C4.4D4), (C2×Dic5).48D6, (C22×D5).24D6, C30.127(C4○D4), C10.45(C4○D12), C6.74(D42D5), D10⋊Dic320C2, C2.24(C20⋊D6), (C2×C60).170C22, (C2×C30).154C23, (C2×Dic3).48D10, C33(Dic5.5D4), C53(C23.11D6), (C22×S3).22D10, C10.74(D42S3), (C6×Dic5).93C22, C2.19(C30.C23), C2.31(D6.D10), (C10×Dic3).93C22, (C2×Dic15).215C22, (C2×C15⋊Q8)⋊12C2, (C5×D6⋊C4)⋊13C2, (C2×C4).183(S3×D5), (C2×C15⋊D4).9C2, (D5×C2×C6).38C22, C22.206(C2×S3×D5), (S3×C2×C10).38C22, (C3×D10⋊C4)⋊13C2, (C2×C6).166(C22×D5), (C2×C10).166(C22×S3), SmallGroup(480,540)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic15.31D4
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — Dic15.31D4
C15C2×C30 — Dic15.31D4
C1C22C2×C4

Generators and relations for Dic15.31D4
 G = < a,b,c,d | a30=c4=1, b2=d2=a15, bab-1=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=a15c-1 >

Subgroups: 812 in 152 conjugacy classes, 46 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C22, C22 [×6], C5, S3, C6 [×3], C6, C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], D5, C10 [×3], C10, Dic3 [×4], C12 [×2], D6 [×3], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×4], C20 [×2], D10 [×3], C2×C10, C2×C10 [×3], Dic6 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30 [×3], C4.4D4, Dic10 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×C10, C4×Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15 [×2], Dic15, C60, C6×D5 [×3], S3×C10 [×3], C2×C30, C4×Dic5, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C23.11D6, C15⋊D4 [×2], C15⋊Q8 [×2], C6×Dic5, C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, Dic5.5D4, D10⋊Dic3, D6⋊Dic5, C3×D10⋊C4, C5×D6⋊C4, C4×Dic15, C2×C15⋊D4, C2×C15⋊Q8, Dic15.31D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C4.4D4, C22×D5, C4○D12, S3×D4, D42S3, S3×D5, C4○D20, D4×D5, D42D5, C23.11D6, C2×S3×D5, Dic5.5D4, D6.D10, C20⋊D6, C30.C23, Dic15.31D4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order122222344444444556666610···1010101010121212121515202020202020202030···3060···60
size111112202221220303030302222220202···212121212442020444444121212124···44···4

60 irreducible representations

dim11111111222222222222444444444
type++++++++++++++++++-++-+-
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10D10C4○D12C4○D20S3×D4D42S3S3×D5D4×D5D42D5C2×S3×D5D6.D10C20⋊D6C30.C23
kernelDic15.31D4D10⋊Dic3D6⋊Dic5C3×D10⋊C4C5×D6⋊C4C4×Dic15C2×C15⋊D4C2×C15⋊Q8D10⋊C4Dic15D6⋊C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C22×S3C10C6C10C10C2×C4C6C6C22C2C2C2
# reps11111111122111422248112222444

Matrix representation of Dic15.31D4 in GL6(𝔽61)

1160000
1170000
0048000
00201400
0000600
0000060
,
3980000
8220000
00465600
00331500
00005235
0000369
,
1100000
0110000
0050000
0005000
00002319
00003038
,
32580000
57290000
0018600
00174300
0000500
00001711

G:=sub<GL(6,GF(61))| [1,1,0,0,0,0,16,17,0,0,0,0,0,0,48,20,0,0,0,0,0,14,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[39,8,0,0,0,0,8,22,0,0,0,0,0,0,46,33,0,0,0,0,56,15,0,0,0,0,0,0,52,36,0,0,0,0,35,9],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,23,30,0,0,0,0,19,38],[32,57,0,0,0,0,58,29,0,0,0,0,0,0,18,17,0,0,0,0,6,43,0,0,0,0,0,0,50,17,0,0,0,0,0,11] >;

Dic15.31D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}._{31}D_4
% in TeX

G:=Group("Dic15.31D4");
// GroupNames label

G:=SmallGroup(480,540);
// by ID

G=gap.SmallGroup(480,540);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=1,b^2=d^2=a^15,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

׿
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