Copied to
clipboard

G = Dic152D4order 480 = 25·3·5

2nd semidirect product of Dic15 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic152D4, (S3×C10)⋊2D4, (C2×D12)⋊5D5, C6.46(D4×D5), C55(Dic3⋊D4), D62(C5⋊D4), C158(C4⋊D4), C10.47(S3×D4), C30.62(C2×D4), (C10×D12)⋊15C2, D6⋊Dic518C2, (C2×C20).230D6, (C2×C12).26D10, D10⋊C417S3, C30.87(C4○D4), C32(Dic5⋊D4), C30.4Q821C2, (C22×D5).14D6, C10.61(C4○D12), C6.30(D42D5), C2.21(C20⋊D6), (C2×C60).323C22, (C2×C30).143C23, (C2×Dic5).118D6, (C22×S3).18D10, C2.17(D125D5), (C6×Dic5).85C22, (C2×Dic15).110C22, (C2×C15⋊D4)⋊4C2, (C2×S3×Dic5)⋊10C2, (C2×C4).59(S3×D5), C2.16(S3×C5⋊D4), C6.37(C2×C5⋊D4), (D5×C2×C6).28C22, C22.195(C2×S3×D5), (S3×C2×C10).33C22, (C3×D10⋊C4)⋊22C2, (C2×C6).155(C22×D5), (C2×C10).155(C22×S3), SmallGroup(480,529)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic152D4
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — Dic152D4
C15C2×C30 — Dic152D4
C1C22C2×C4

Generators and relations for Dic152D4
 G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, ac=ca, dad=a11, cbc-1=dbd=a15b, dcd=c-1 >

Subgroups: 1004 in 188 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3 [×3], C6 [×3], C6, C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5, C10 [×3], C10 [×3], Dic3 [×3], C12 [×2], D6 [×2], D6 [×5], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×4], C20, D10 [×3], C2×C10, C2×C10 [×7], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C2×C12, C22×S3 [×2], C22×C6, C5×S3 [×3], C3×D5, C30 [×3], C4⋊D4, C2×Dic5, C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, C6×D5 [×3], S3×C10 [×2], S3×C10 [×5], C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4 [×2], D4×C10, Dic3⋊D4, S3×Dic5 [×2], C15⋊D4 [×4], C6×Dic5, C5×D12 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10 [×2], Dic5⋊D4, D6⋊Dic5, C3×D10⋊C4, C30.4Q8, C2×S3×Dic5, C2×C15⋊D4 [×2], C10×D12, Dic152D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C22×S3, C4⋊D4, C5⋊D4 [×2], C22×D5, C4○D12, S3×D4 [×2], S3×D5, D4×D5, D42D5, C2×C5⋊D4, Dic3⋊D4, C2×S3×D5, Dic5⋊D4, D125D5, C20⋊D6, S3×C5⋊D4, Dic152D4

Smallest permutation representation of Dic152D4
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 53 16 38)(2 52 17 37)(3 51 18 36)(4 50 19 35)(5 49 20 34)(6 48 21 33)(7 47 22 32)(8 46 23 31)(9 45 24 60)(10 44 25 59)(11 43 26 58)(12 42 27 57)(13 41 28 56)(14 40 29 55)(15 39 30 54)(61 160 76 175)(62 159 77 174)(63 158 78 173)(64 157 79 172)(65 156 80 171)(66 155 81 170)(67 154 82 169)(68 153 83 168)(69 152 84 167)(70 151 85 166)(71 180 86 165)(72 179 87 164)(73 178 88 163)(74 177 89 162)(75 176 90 161)(91 147 106 132)(92 146 107 131)(93 145 108 130)(94 144 109 129)(95 143 110 128)(96 142 111 127)(97 141 112 126)(98 140 113 125)(99 139 114 124)(100 138 115 123)(101 137 116 122)(102 136 117 121)(103 135 118 150)(104 134 119 149)(105 133 120 148)(181 227 196 212)(182 226 197 211)(183 225 198 240)(184 224 199 239)(185 223 200 238)(186 222 201 237)(187 221 202 236)(188 220 203 235)(189 219 204 234)(190 218 205 233)(191 217 206 232)(192 216 207 231)(193 215 208 230)(194 214 209 229)(195 213 210 228)
(1 195 74 129)(2 196 75 130)(3 197 76 131)(4 198 77 132)(5 199 78 133)(6 200 79 134)(7 201 80 135)(8 202 81 136)(9 203 82 137)(10 204 83 138)(11 205 84 139)(12 206 85 140)(13 207 86 141)(14 208 87 142)(15 209 88 143)(16 210 89 144)(17 181 90 145)(18 182 61 146)(19 183 62 147)(20 184 63 148)(21 185 64 149)(22 186 65 150)(23 187 66 121)(24 188 67 122)(25 189 68 123)(26 190 69 124)(27 191 70 125)(28 192 71 126)(29 193 72 127)(30 194 73 128)(31 236 155 117)(32 237 156 118)(33 238 157 119)(34 239 158 120)(35 240 159 91)(36 211 160 92)(37 212 161 93)(38 213 162 94)(39 214 163 95)(40 215 164 96)(41 216 165 97)(42 217 166 98)(43 218 167 99)(44 219 168 100)(45 220 169 101)(46 221 170 102)(47 222 171 103)(48 223 172 104)(49 224 173 105)(50 225 174 106)(51 226 175 107)(52 227 176 108)(53 228 177 109)(54 229 178 110)(55 230 179 111)(56 231 180 112)(57 232 151 113)(58 233 152 114)(59 234 153 115)(60 235 154 116)
(1 129)(2 140)(3 121)(4 132)(5 143)(6 124)(7 135)(8 146)(9 127)(10 138)(11 149)(12 130)(13 141)(14 122)(15 133)(16 144)(17 125)(18 136)(19 147)(20 128)(21 139)(22 150)(23 131)(24 142)(25 123)(26 134)(27 145)(28 126)(29 137)(30 148)(31 107)(32 118)(33 99)(34 110)(35 91)(36 102)(37 113)(38 94)(39 105)(40 116)(41 97)(42 108)(43 119)(44 100)(45 111)(46 92)(47 103)(48 114)(49 95)(50 106)(51 117)(52 98)(53 109)(54 120)(55 101)(56 112)(57 93)(58 104)(59 115)(60 96)(61 202)(62 183)(63 194)(64 205)(65 186)(66 197)(67 208)(68 189)(69 200)(70 181)(71 192)(72 203)(73 184)(74 195)(75 206)(76 187)(77 198)(78 209)(79 190)(80 201)(81 182)(82 193)(83 204)(84 185)(85 196)(86 207)(87 188)(88 199)(89 210)(90 191)(151 212)(152 223)(153 234)(154 215)(155 226)(156 237)(157 218)(158 229)(159 240)(160 221)(161 232)(162 213)(163 224)(164 235)(165 216)(166 227)(167 238)(168 219)(169 230)(170 211)(171 222)(172 233)(173 214)(174 225)(175 236)(176 217)(177 228)(178 239)(179 220)(180 231)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E10A···10F10G···10N12A12B12C12D15A15B20A20B20C20D30A···30F60A···60H
order122222223444444556666610···1010···101212121215152020202030···3060···60
size11116612202410103030602222220202···212···124420204444444···44···4

60 irreducible representations

dim111111122222222222244444444
type+++++++++++++++++++-+-
imageC1C2C2C2C2C2C2S3D4D4D5D6D6D6C4○D4D10D10C5⋊D4C4○D12S3×D4S3×D5D4×D5D42D5C2×S3×D5D125D5C20⋊D6S3×C5⋊D4
kernelDic152D4D6⋊Dic5C3×D10⋊C4C30.4Q8C2×S3×Dic5C2×C15⋊D4C10×D12D10⋊C4Dic15S3×C10C2×D12C2×Dic5C2×C20C22×D5C30C2×C12C22×S3D6C10C10C2×C4C6C6C22C2C2C2
# reps111112112221112248422222444

Matrix representation of Dic152D4 in GL6(𝔽61)

0600000
110000
0060100
00164400
000010
000001
,
1100000
50500000
0001800
0017000
000010
000001
,
38150000
46230000
001000
000100
00001520
0000746
,
38150000
38230000
001000
000100
00001520
0000146

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,60,16,0,0,0,0,1,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,50,0,0,0,0,0,50,0,0,0,0,0,0,0,17,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,46,0,0,0,0,15,23,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,7,0,0,0,0,20,46],[38,38,0,0,0,0,15,23,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,1,0,0,0,0,20,46] >;

Dic152D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_2D_4
% in TeX

G:=Group("Dic15:2D4");
// GroupNames label

G:=SmallGroup(480,529);
// by ID

G=gap.SmallGroup(480,529);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,219,100,1356,18822]);
// Polycyclic

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

׿
×
𝔽