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G = D307D4order 480 = 25·3·5

7th semidirect product of D30 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D307D4, C6.85(D4×D5), C56(Dic3⋊D4), C32(C202D4), (C5×Dic3)⋊4D4, C10.87(S3×D4), C23.D58S3, C1519(C4⋊D4), D6⋊Dic532C2, C30.237(C2×D4), C23.21(S3×D5), Dic33(C5⋊D4), C6.Dic1035C2, (C2×Dic5).62D6, (C22×C10).51D6, (C22×C6).36D10, C30.151(C4○D4), C10.84(C4○D12), C6.57(D42D5), (C2×C30).199C23, (C22×S3).28D10, C2.39(D10⋊D6), (C2×Dic3).123D10, (C22×C30).61C22, C2.27(Dic3.D10), (C6×Dic5).115C22, (C22×D15).65C22, (C10×Dic3).115C22, (C2×Dic15).138C22, (C2×C3⋊D4)⋊2D5, (C10×C3⋊D4)⋊2C2, C6.64(C2×C5⋊D4), C2.39(S3×C5⋊D4), (C2×C157D4)⋊14C2, (C2×C5⋊D12)⋊11C2, C22.231(C2×S3×D5), (C2×D30.C2)⋊16C2, (S3×C2×C10).50C22, (C3×C23.D5)⋊10C2, (C2×C6).211(C22×D5), (C2×C10).211(C22×S3), SmallGroup(480,633)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D307D4
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D307D4
C15C2×C30 — D307D4
C1C22C23

Generators and relations for D307D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=a10b, dbd=a25b, dcd=c-1 >

Subgroups: 1068 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 [×6], D4 [×6], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], Dic3, C12 [×2], D6 [×7], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×6], C4×S3 [×2], D12 [×2], C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×S3, C22×C6, C5×S3, D15 [×2], C30 [×3], C30, C4⋊D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, S3×C10 [×3], D30 [×2], D30 [×2], C2×C30, C2×C30 [×3], C4⋊Dic5, C23.D5, C23.D5, C2×C4×D5, C2×C5⋊D4 [×2], D4×C10, Dic3⋊D4, D30.C2 [×2], C5⋊D12 [×2], C6×Dic5 [×2], C10×Dic3, C5×C3⋊D4 [×2], C2×Dic15, C157D4 [×2], S3×C2×C10, C22×D15, C22×C30, C202D4, D6⋊Dic5, C6.Dic10, C3×C23.D5, C2×D30.C2, C2×C5⋊D12, C10×C3⋊D4, C2×C157D4, D307D4
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, C202D4, Dic3.D10, S3×C5⋊D4, D10⋊D6, D307D4

Smallest permutation representation of D307D4
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)
(1 211)(2 240)(3 239)(4 238)(5 237)(6 236)(7 235)(8 234)(9 233)(10 232)(11 231)(12 230)(13 229)(14 228)(15 227)(16 226)(17 225)(18 224)(19 223)(20 222)(21 221)(22 220)(23 219)(24 218)(25 217)(26 216)(27 215)(28 214)(29 213)(30 212)(31 167)(32 166)(33 165)(34 164)(35 163)(36 162)(37 161)(38 160)(39 159)(40 158)(41 157)(42 156)(43 155)(44 154)(45 153)(46 152)(47 151)(48 180)(49 179)(50 178)(51 177)(52 176)(53 175)(54 174)(55 173)(56 172)(57 171)(58 170)(59 169)(60 168)(61 129)(62 128)(63 127)(64 126)(65 125)(66 124)(67 123)(68 122)(69 121)(70 150)(71 149)(72 148)(73 147)(74 146)(75 145)(76 144)(77 143)(78 142)(79 141)(80 140)(81 139)(82 138)(83 137)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 130)(91 184)(92 183)(93 182)(94 181)(95 210)(96 209)(97 208)(98 207)(99 206)(100 205)(101 204)(102 203)(103 202)(104 201)(105 200)(106 199)(107 198)(108 197)(109 196)(110 195)(111 194)(112 193)(113 192)(114 191)(115 190)(116 189)(117 188)(118 187)(119 186)(120 185)
(1 166 212 33)(2 177 213 44)(3 158 214 55)(4 169 215 36)(5 180 216 47)(6 161 217 58)(7 172 218 39)(8 153 219 50)(9 164 220 31)(10 175 221 42)(11 156 222 53)(12 167 223 34)(13 178 224 45)(14 159 225 56)(15 170 226 37)(16 151 227 48)(17 162 228 59)(18 173 229 40)(19 154 230 51)(20 165 231 32)(21 176 232 43)(22 157 233 54)(23 168 234 35)(24 179 235 46)(25 160 236 57)(26 171 237 38)(27 152 238 49)(28 163 239 60)(29 174 240 41)(30 155 211 52)(61 187 143 112)(62 198 144 93)(63 209 145 104)(64 190 146 115)(65 201 147 96)(66 182 148 107)(67 193 149 118)(68 204 150 99)(69 185 121 110)(70 196 122 91)(71 207 123 102)(72 188 124 113)(73 199 125 94)(74 210 126 105)(75 191 127 116)(76 202 128 97)(77 183 129 108)(78 194 130 119)(79 205 131 100)(80 186 132 111)(81 197 133 92)(82 208 134 103)(83 189 135 114)(84 200 136 95)(85 181 137 106)(86 192 138 117)(87 203 139 98)(88 184 140 109)(89 195 141 120)(90 206 142 101)
(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)(31 116)(32 97)(33 108)(34 119)(35 100)(36 111)(37 92)(38 103)(39 114)(40 95)(41 106)(42 117)(43 98)(44 109)(45 120)(46 101)(47 112)(48 93)(49 104)(50 115)(51 96)(52 107)(53 118)(54 99)(55 110)(56 91)(57 102)(58 113)(59 94)(60 105)(61 216)(62 227)(63 238)(64 219)(65 230)(66 211)(67 222)(68 233)(69 214)(70 225)(71 236)(72 217)(73 228)(74 239)(75 220)(76 231)(77 212)(78 223)(79 234)(80 215)(81 226)(82 237)(83 218)(84 229)(85 240)(86 221)(87 232)(88 213)(89 224)(90 235)(151 198)(152 209)(153 190)(154 201)(155 182)(156 193)(157 204)(158 185)(159 196)(160 207)(161 188)(162 199)(163 210)(164 191)(165 202)(166 183)(167 194)(168 205)(169 186)(170 197)(171 208)(172 189)(173 200)(174 181)(175 192)(176 203)(177 184)(178 195)(179 206)(180 187)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E10A···10F10G10H10I10J10K10L10M10N12A12B12C12D15A15B20A20B20C20D30A···30N
order122222223444444556666610···1010101010101010101212121215152020202030···30
size111141230302661010206022222442···24444121212122020202044121212124···4

60 irreducible representations

dim1111111122222222222244444444
type++++++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10D10C5⋊D4C4○D12S3×D4S3×D5D4×D5D42D5C2×S3×D5Dic3.D10S3×C5⋊D4D10⋊D6
kernelD307D4D6⋊Dic5C6.Dic10C3×C23.D5C2×D30.C2C2×C5⋊D12C10×C3⋊D4C2×C157D4C23.D5C5×Dic3D30C2×C3⋊D4C2×Dic5C22×C10C30C2×Dic3C22×S3C22×C6Dic3C10C10C23C6C6C22C2C2C2
# reps1111111112222122228422222444

Matrix representation of D307D4 in GL6(𝔽61)

6000000
0600000
00186000
00196000
0000160
000010
,
6000000
4110000
0001700
0018000
0000600
0000601
,
100000
010000
0060000
0006000
0000050
0000500
,
160000
0600000
001000
000100
00001523
00003846

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,18,19,0,0,0,0,60,60,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[60,41,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,60,60,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,50,0,0,0,0,50,0],[1,0,0,0,0,0,6,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,38,0,0,0,0,23,46] >;

D307D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_7D_4
% in TeX

G:=Group("D30:7D4");
// GroupNames label

G:=SmallGroup(480,633);
// by ID

G=gap.SmallGroup(480,633);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=a^10*b,d*b*d=a^25*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽