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G = D308Q8order 480 = 25·3·5

4th semidirect product of D30 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D308Q8, Dic15.39D4, C6.7(Q8×D5), C6.55(D4×D5), C10.7(S3×Q8), C51(D6⋊Q8), C10.57(S3×D4), C157(C22⋊Q8), C30.26(C2×Q8), Dic3⋊C411D5, C31(D10⋊Q8), (C2×C20).185D6, C30.119(C2×D4), D304C4.4C2, C6.28(C4○D20), C30.42(C4○D4), C10.D411S3, (C2×C12).183D10, C2.10(D15⋊Q8), (C2×C30).67C23, Dic155C410C2, (C2×Dic5).19D6, C10.31(C4○D12), (C2×C60).161C22, (C2×Dic3).20D10, C2.11(D10⋊D6), (C6×Dic5).38C22, C2.20(D6.D10), (C10×Dic3).39C22, (C22×D15).97C22, (C2×Dic15).191C22, (C2×C15⋊Q8)⋊5C2, (C2×C4×D15).7C2, (C2×C4).174(S3×D5), C22.153(C2×S3×D5), (C5×Dic3⋊C4)⋊11C2, (C2×C6).79(C22×D5), (C3×C10.D4)⋊11C2, (C2×C10).79(C22×S3), SmallGroup(480,453)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D308Q8
C1C5C15C30C2×C30C6×Dic5Dic155C4 — D308Q8
C15C2×C30 — D308Q8
C1C22C2×C4

Generators and relations for D308Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a19, cbc-1=a28b, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 844 in 148 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×4], C12 [×3], D6 [×4], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20 [×3], D10 [×4], C2×C10, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, Dic3⋊C4, Dic3⋊C4, D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60, D30 [×2], D30 [×2], C2×C30, C10.D4, C10.D4, D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, D6⋊Q8, C15⋊Q8 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C4×D15 [×2], C2×Dic15, C2×C60, C22×D15, D10⋊Q8, D304C4 [×2], Dic155C4, C3×C10.D4, C5×Dic3⋊C4, C2×C15⋊Q8, C2×C4×D15, D308Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C22×S3, C22⋊Q8, C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C4○D20, D4×D5, Q8×D5, D6⋊Q8, C2×S3×D5, D10⋊Q8, D15⋊Q8, D6.D10, D10⋊D6, D308Q8

Smallest permutation representation of D308Q8
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)
(1 198)(2 197)(3 196)(4 195)(5 194)(6 193)(7 192)(8 191)(9 190)(10 189)(11 188)(12 187)(13 186)(14 185)(15 184)(16 183)(17 182)(18 181)(19 210)(20 209)(21 208)(22 207)(23 206)(24 205)(25 204)(26 203)(27 202)(28 201)(29 200)(30 199)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 120)(38 119)(39 118)(40 117)(41 116)(42 115)(43 114)(44 113)(45 112)(46 111)(47 110)(48 109)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 159)(62 158)(63 157)(64 156)(65 155)(66 154)(67 153)(68 152)(69 151)(70 180)(71 179)(72 178)(73 177)(74 176)(75 175)(76 174)(77 173)(78 172)(79 171)(80 170)(81 169)(82 168)(83 167)(84 166)(85 165)(86 164)(87 163)(88 162)(89 161)(90 160)(121 228)(122 227)(123 226)(124 225)(125 224)(126 223)(127 222)(128 221)(129 220)(130 219)(131 218)(132 217)(133 216)(134 215)(135 214)(136 213)(137 212)(138 211)(139 240)(140 239)(141 238)(142 237)(143 236)(144 235)(145 234)(146 233)(147 232)(148 231)(149 230)(150 229)
(1 228 184 137)(2 227 185 136)(3 226 186 135)(4 225 187 134)(5 224 188 133)(6 223 189 132)(7 222 190 131)(8 221 191 130)(9 220 192 129)(10 219 193 128)(11 218 194 127)(12 217 195 126)(13 216 196 125)(14 215 197 124)(15 214 198 123)(16 213 199 122)(17 212 200 121)(18 211 201 150)(19 240 202 149)(20 239 203 148)(21 238 204 147)(22 237 205 146)(23 236 206 145)(24 235 207 144)(25 234 208 143)(26 233 209 142)(27 232 210 141)(28 231 181 140)(29 230 182 139)(30 229 183 138)(31 71 111 166)(32 70 112 165)(33 69 113 164)(34 68 114 163)(35 67 115 162)(36 66 116 161)(37 65 117 160)(38 64 118 159)(39 63 119 158)(40 62 120 157)(41 61 91 156)(42 90 92 155)(43 89 93 154)(44 88 94 153)(45 87 95 152)(46 86 96 151)(47 85 97 180)(48 84 98 179)(49 83 99 178)(50 82 100 177)(51 81 101 176)(52 80 102 175)(53 79 103 174)(54 78 104 173)(55 77 105 172)(56 76 106 171)(57 75 107 170)(58 74 108 169)(59 73 109 168)(60 72 110 167)
(1 158 184 63)(2 177 185 82)(3 166 186 71)(4 155 187 90)(5 174 188 79)(6 163 189 68)(7 152 190 87)(8 171 191 76)(9 160 192 65)(10 179 193 84)(11 168 194 73)(12 157 195 62)(13 176 196 81)(14 165 197 70)(15 154 198 89)(16 173 199 78)(17 162 200 67)(18 151 201 86)(19 170 202 75)(20 159 203 64)(21 178 204 83)(22 167 205 72)(23 156 206 61)(24 175 207 80)(25 164 208 69)(26 153 209 88)(27 172 210 77)(28 161 181 66)(29 180 182 85)(30 169 183 74)(31 226 111 135)(32 215 112 124)(33 234 113 143)(34 223 114 132)(35 212 115 121)(36 231 116 140)(37 220 117 129)(38 239 118 148)(39 228 119 137)(40 217 120 126)(41 236 91 145)(42 225 92 134)(43 214 93 123)(44 233 94 142)(45 222 95 131)(46 211 96 150)(47 230 97 139)(48 219 98 128)(49 238 99 147)(50 227 100 136)(51 216 101 125)(52 235 102 144)(53 224 103 133)(54 213 104 122)(55 232 105 141)(56 221 106 130)(57 240 107 149)(58 229 108 138)(59 218 109 127)(60 237 110 146)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222223444444445566610···1012121212121215152020202020···2030···3060···60
size11113030222121220203030222222···2442020202044444412···124···44···4

60 irreducible representations

dim111111122222222222444444444
type+++++++++-++++++-++-++
imageC1C2C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10C4○D12C4○D20S3×D4S3×Q8S3×D5D4×D5Q8×D5C2×S3×D5D15⋊Q8D6.D10D10⋊D6
kernelD308Q8D304C4Dic155C4C3×C10.D4C5×Dic3⋊C4C2×C15⋊Q8C2×C4×D15C10.D4Dic15D30Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C10C6C10C10C2×C4C6C6C22C2C2C2
# reps121111112222124248112222444

Matrix representation of D308Q8 in GL6(𝔽61)

010000
60180000
0060100
0060000
000010
000001
,
4310000
43180000
0060000
0060100
0000600
0000060
,
15460000
11460000
000100
001000
00001456
00001547
,
880000
30530000
001000
000100
00004144
00002020

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,18,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[43,43,0,0,0,0,1,18,0,0,0,0,0,0,60,60,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[15,11,0,0,0,0,46,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,14,15,0,0,0,0,56,47],[8,30,0,0,0,0,8,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,41,20,0,0,0,0,44,20] >;

D308Q8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_8Q_8
% in TeX

G:=Group("D30:8Q8");
// GroupNames label

G:=SmallGroup(480,453);
// by ID

G=gap.SmallGroup(480,453);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,422,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^19,c*b*c^-1=a^28*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽