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G = Dic15.19D4  order 480 = 25·3·5

19th non-split extension by Dic15 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic15.19D4, C6.83(D4×D5), C10.84(S3×D4), C23.D5⋊4S3, C30.212(C2×D4), C6.D4⋊3D5, D30⋊4C4⋊25C2, C23.12(S3×D5), C6.74(C4○D20), C15⋊14(C4.4D4), (C2×Dic5).52D6, (C22×C10).33D6, (C22×C6).18D10, (Dic3×Dic5)⋊26C2, C30.129(C4○D4), C10.75(C4○D12), C6.49(D4⋊2D5), (C2×C30).168C23, C5⋊4(C23.11D6), C3⋊4(Dic5.5D4), C2.36(D10⋊D6), C10.48(D4⋊2S3), (C2×Dic3).116D10, (C22×C30).30C22, (C6×Dic5).98C22, C2.21(Dic3.D10), C2.20(Dic5.D6), (C10×Dic3).98C22, (C22×D15).58C22, (C2×Dic15).119C22, (C2×C15⋊Q8)⋊14C2, (C2×C15⋊7D4).8C2, (C3×C23.D5)⋊4C2, C22.216(C2×S3×D5), (C5×C6.D4)⋊3C2, (C2×C6).180(C22×D5), (C2×C10).180(C22×S3), SmallGroup(480,602)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C30 — Dic15.19D4
Chief series
C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic15.19D4
Lower central
C15 — C2×C30 — Dic15.19D4
Upper central
C1 — C22 — C23

Generators and relations for Dic15.19D4
 G = < a,b,c,d | a30=c4=1, b2=d2=a15, bab-1=a-1, cac-1=dad-1=a19, bc=cb, dbd-1=a15b, dcd-1=a15c-1 >

Subgroups: 876 in 152 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, Q8, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C4.4D4, Dic10, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, D30, C2×C30, C2×C30, C4×Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, C23.11D6, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C15⋊7D4, C22×D15, C22×C30, Dic5.5D4, Dic3×Dic5, D30⋊4C4, C3×C23.D5, C5×C6.D4, C2×C15⋊Q8, C2×C15⋊7D4, Dic15.19D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, C4.4D4, C22×D5, C4○D12, S3×D4, D4⋊2S3, S3×D5, C4○D20, D4×D5, D4⋊2D5, C23.11D6, C2×S3×D5, Dic5.5D4, Dic5.D6, Dic3.D10, D10⋊D6, Dic15.19D4

Smallest permutation representation of Dic15.19D4
►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)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A···20H30A···30N
order122222344444444556666610···101010101012121212151520···2030···30
size111146026610101220303022222442···24444202020204412···124···4

60 irreducible representations

dim11111112222222222444444444
type+++++++++++++++-++-++
imageC1C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10C4○D12C4○D20S3×D4D4⋊2S3S3×D5D4×D5D4⋊2D5C2×S3×D5Dic5.D6Dic3.D10D10⋊D6
kernelDic15.19D4Dic3×Dic5D30⋊4C4C3×C23.D5C5×C6.D4C2×C15⋊Q8C2×C15⋊7D4C23.D5Dic15C6.D4C2×Dic5C22×C10C30C2×Dic3C22×C6C10C6C10C10C23C6C6C22C2C2C2
# reps11211111222144248112222444

Matrix representation of Dic15.19D4 ►in GL8(𝔽61)

600000000
060000000
006010000
006000000
000018100
0000426000
000000600
000000060
,
609000000
541000000
000600000
006000000
0000181700
0000424300
0000005360
00000048
,
1123000000
1650000000
00100000
00010000
0000434400
0000191800
000000110
000000011
,
500000000
4511000000
006000000
000600000
0000181700
0000424300
0000003450
0000002227

G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,42,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[60,54,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,18,42,0,0,0,0,0,0,17,43,0,0,0,0,0,0,0,0,53,4,0,0,0,0,0,0,60,8],[11,16,0,0,0,0,0,0,23,50,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,43,19,0,0,0,0,0,0,44,18,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[50,45,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,18,42,0,0,0,0,0,0,17,43,0,0,0,0,0,0,0,0,34,22,0,0,0,0,0,0,50,27] >;
 

Dic15.19D4 in GAP, Magma, Sage, TeX 

{\rm Dic}_{15}._{19}D_4
 
% in TeX
 
G:=Group("Dic15.19D4");
 
// GroupNames label
 
G:=SmallGroup(480,602);
 
// by ID
 
G=gap.SmallGroup(480,602);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,219,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^30=c^4=1,b^2=d^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^19,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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