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G = D306Q8order 480 = 25·3·5

2nd semidirect product of D30 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D306Q8, C4.13D60, C60.11D4, C12.23D20, C20.23D12, C4⋊C45D15, C6.43(Q8×D5), C2.7(Q8×D15), C605C412C2, C53(C4.D12), (C2×C4).44D30, C6.36(C2×D20), C2.10(C2×D60), C10.43(S3×Q8), C30.96(C2×Q8), C33(D102Q8), (C2×C12).40D10, (C2×C20).139D6, C30.264(C2×D4), C10.37(C2×D12), C1529(C22⋊Q8), (C2×Dic30)⋊10C2, D303C4.3C2, (C2×C60).21C22, C30.220(C4○D4), C6.99(D42D5), (C2×C30).294C23, C2.13(D42D15), C10.99(D42S3), C22.52(C22×D15), (C2×Dic15).12C22, (C22×D15).84C22, (C5×C4⋊C4)⋊8S3, (C3×C4⋊C4)⋊8D5, (C15×C4⋊C4)⋊8C2, (C2×C4×D15).2C2, (C2×C6).290(C22×D5), (C2×C10).289(C22×S3), SmallGroup(480,862)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D306Q8
C1C5C15C30C2×C30C22×D15C2×C4×D15 — D306Q8
C15C2×C30 — D306Q8
C1C22C4⋊C4

Generators and relations for D306Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a13b, bd=db, dcd-1=c-1 >

Subgroups: 900 in 148 conjugacy classes, 57 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], C2×C10, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C22×S3, D15 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×C20, C2×C20 [×2], C22×D5, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, Dic15 [×3], C60 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C4.D12, Dic30 [×2], C4×D15 [×2], C2×Dic15, C2×Dic15 [×2], C2×C60, C2×C60 [×2], C22×D15, D102Q8, C605C4 [×2], D303C4 [×2], C15×C4⋊C4, C2×Dic30, C2×C4×D15, D306Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], D12 [×2], C22×S3, D15, C22⋊Q8, D20 [×2], C22×D5, C2×D12, D42S3, S3×Q8, D30 [×3], C2×D20, D42D5, Q8×D5, C4.D12, D60 [×2], C22×D15, D102Q8, C2×D60, D42D15, Q8×D15, D306Q8

Smallest permutation representation of D306Q8
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)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 150)(15 149)(16 148)(17 147)(18 146)(19 145)(20 144)(21 143)(22 142)(23 141)(24 140)(25 139)(26 138)(27 137)(28 136)(29 135)(30 134)(31 202)(32 201)(33 200)(34 199)(35 198)(36 197)(37 196)(38 195)(39 194)(40 193)(41 192)(42 191)(43 190)(44 189)(45 188)(46 187)(47 186)(48 185)(49 184)(50 183)(51 182)(52 181)(53 210)(54 209)(55 208)(56 207)(57 206)(58 205)(59 204)(60 203)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 120)(73 119)(74 118)(75 117)(76 116)(77 115)(78 114)(79 113)(80 112)(81 111)(82 110)(83 109)(84 108)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(151 213)(152 212)(153 211)(154 240)(155 239)(156 238)(157 237)(158 236)(159 235)(160 234)(161 233)(162 232)(163 231)(164 230)(165 229)(166 228)(167 227)(168 226)(169 225)(170 224)(171 223)(172 222)(173 221)(174 220)(175 219)(176 218)(177 217)(178 216)(179 215)(180 214)
(1 40 134 209)(2 39 135 208)(3 38 136 207)(4 37 137 206)(5 36 138 205)(6 35 139 204)(7 34 140 203)(8 33 141 202)(9 32 142 201)(10 31 143 200)(11 60 144 199)(12 59 145 198)(13 58 146 197)(14 57 147 196)(15 56 148 195)(16 55 149 194)(17 54 150 193)(18 53 121 192)(19 52 122 191)(20 51 123 190)(21 50 124 189)(22 49 125 188)(23 48 126 187)(24 47 127 186)(25 46 128 185)(26 45 129 184)(27 44 130 183)(28 43 131 182)(29 42 132 181)(30 41 133 210)(61 218 95 169)(62 217 96 168)(63 216 97 167)(64 215 98 166)(65 214 99 165)(66 213 100 164)(67 212 101 163)(68 211 102 162)(69 240 103 161)(70 239 104 160)(71 238 105 159)(72 237 106 158)(73 236 107 157)(74 235 108 156)(75 234 109 155)(76 233 110 154)(77 232 111 153)(78 231 112 152)(79 230 113 151)(80 229 114 180)(81 228 115 179)(82 227 116 178)(83 226 117 177)(84 225 118 176)(85 224 119 175)(86 223 120 174)(87 222 91 173)(88 221 92 172)(89 220 93 171)(90 219 94 170)
(1 173 134 222)(2 174 135 223)(3 175 136 224)(4 176 137 225)(5 177 138 226)(6 178 139 227)(7 179 140 228)(8 180 141 229)(9 151 142 230)(10 152 143 231)(11 153 144 232)(12 154 145 233)(13 155 146 234)(14 156 147 235)(15 157 148 236)(16 158 149 237)(17 159 150 238)(18 160 121 239)(19 161 122 240)(20 162 123 211)(21 163 124 212)(22 164 125 213)(23 165 126 214)(24 166 127 215)(25 167 128 216)(26 168 129 217)(27 169 130 218)(28 170 131 219)(29 171 132 220)(30 172 133 221)(31 112 200 78)(32 113 201 79)(33 114 202 80)(34 115 203 81)(35 116 204 82)(36 117 205 83)(37 118 206 84)(38 119 207 85)(39 120 208 86)(40 91 209 87)(41 92 210 88)(42 93 181 89)(43 94 182 90)(44 95 183 61)(45 96 184 62)(46 97 185 63)(47 98 186 64)(48 99 187 65)(49 100 188 66)(50 101 189 67)(51 102 190 68)(52 103 191 69)(53 104 192 70)(54 105 193 71)(55 106 194 72)(56 107 195 73)(57 108 196 74)(58 109 197 75)(59 110 198 76)(60 111 199 77)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222223444444445566610···1012···121515151520···2030···3060···60
size111130302224430306060222222···24···422224···42···24···4

84 irreducible representations

dim111111222222222222444444
type++++++++-++++++++------
imageC1C2C2C2C2C2S3D4Q8D5D6C4○D4D10D12D15D20D30D60D42S3S3×Q8D42D5Q8×D5D42D15Q8×D15
kernelD306Q8C605C4D303C4C15×C4⋊C4C2×Dic30C2×C4×D15C5×C4⋊C4C60D30C3×C4⋊C4C2×C20C30C2×C12C20C4⋊C4C12C2×C4C4C10C10C6C6C2C2
# reps12211112223264481216112244

Matrix representation of D306Q8 in GL6(𝔽61)

5690000
52160000
00581500
0052400
0000600
0000060
,
1810000
43430000
00581500
0032300
000010
00006060
,
8450000
23530000
00394200
00192200
00006059
000011
,
100000
010000
0060000
0006000
0000500
00001111

G:=sub<GL(6,GF(61))| [56,52,0,0,0,0,9,16,0,0,0,0,0,0,58,52,0,0,0,0,15,4,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,43,0,0,0,0,1,43,0,0,0,0,0,0,58,32,0,0,0,0,15,3,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[8,23,0,0,0,0,45,53,0,0,0,0,0,0,39,19,0,0,0,0,42,22,0,0,0,0,0,0,60,1,0,0,0,0,59,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,11,0,0,0,0,0,11] >;

D306Q8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_6Q_8
% in TeX

G:=Group("D30:6Q8");
// GroupNames label

G:=SmallGroup(480,862);
// by ID

G=gap.SmallGroup(480,862);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,142,2693,18822]);
// 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,a*d=d*a,c*b*c^-1=a^13*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽