Copied to
clipboard

G = D30.2Q8order 480 = 25·3·5

2nd non-split extension by D30 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.2Q8, D30.26D4, C208(C4×S3), C124(C4×D5), C6022(C2×C4), (C4×D15)⋊9C4, C6.43(D4×D5), D153(C4⋊C4), C6.17(Q8×D5), C4⋊Dic514S3, C4⋊Dic314D5, C10.44(S3×D4), C30.55(C2×D4), C10.17(S3×Q8), C30.51(C2×Q8), C43(D30.C2), D30.41(C2×C4), (C2×C20).128D6, C2.5(D15⋊Q8), Dic1522(C2×C4), (C2×C12).129D10, C2.3(C20⋊D6), Dic155C423C2, (C2×C60).202C22, C30.132(C22×C4), (C2×C30).127C23, (C2×Dic5).114D6, (C2×Dic3).108D10, (C6×Dic5).79C22, (C10×Dic3).80C22, (C2×Dic15).208C22, (C22×D15).105C22, C53(S3×C4⋊C4), C32(D5×C4⋊C4), C1510(C2×C4⋊C4), C6.50(C2×C4×D5), C10.82(S3×C2×C4), (C2×C4×D15).16C2, C22.60(C2×S3×D5), (C5×C4⋊Dic3)⋊11C2, (C3×C4⋊Dic5)⋊11C2, (C2×C4).212(S3×D5), (C2×D30.C2).8C2, C2.14(C2×D30.C2), (C2×C6).139(C22×D5), (C2×C10).139(C22×S3), SmallGroup(480,513)

Series: Derived Chief Lower central Upper central

C1C30 — D30.2Q8
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D30.2Q8
C15C30 — D30.2Q8
C1C22C2×C4

Generators and relations for D30.2Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=a15c2, bab=a-1, ac=ca, dad-1=a19, bc=cb, dbd-1=a18b, dcd-1=c-1 >

Subgroups: 940 in 184 conjugacy classes, 68 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×13], C23, D5 [×4], C10 [×3], Dic3 [×4], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×4], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4×S3 [×8], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15 [×4], C30 [×3], C2×C4⋊C4, C4×D5 [×8], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4 [×3], C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], D30 [×6], C2×C30, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C2×C4×D5 [×3], S3×C4⋊C4, D30.C2 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, D5×C4⋊C4, Dic155C4 [×2], C3×C4⋊Dic5, C5×C4⋊Dic3, C2×D30.C2 [×2], C2×C4×D15, D30.2Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C4×S3 [×2], C22×S3, C2×C4⋊C4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4, S3×Q8, S3×D5, C2×C4×D5, D4×D5, Q8×D5, S3×C4⋊C4, D30.C2 [×2], C2×S3×D5, D5×C4⋊C4, D15⋊Q8, C20⋊D6, C2×D30.C2, D30.2Q8

Smallest permutation representation of D30.2Q8
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(53 60)(54 59)(55 58)(56 57)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(141 150)(142 149)(143 148)(144 147)(145 146)(151 170)(152 169)(153 168)(154 167)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)(171 180)(172 179)(173 178)(174 177)(175 176)(181 190)(182 189)(183 188)(184 187)(185 186)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(211 216)(212 215)(213 214)(217 240)(218 239)(219 238)(220 237)(221 236)(222 235)(223 234)(224 233)(225 232)(226 231)(227 230)(228 229)
(1 57 176 76)(2 58 177 77)(3 59 178 78)(4 60 179 79)(5 31 180 80)(6 32 151 81)(7 33 152 82)(8 34 153 83)(9 35 154 84)(10 36 155 85)(11 37 156 86)(12 38 157 87)(13 39 158 88)(14 40 159 89)(15 41 160 90)(16 42 161 61)(17 43 162 62)(18 44 163 63)(19 45 164 64)(20 46 165 65)(21 47 166 66)(22 48 167 67)(23 49 168 68)(24 50 169 69)(25 51 170 70)(26 52 171 71)(27 53 172 72)(28 54 173 73)(29 55 174 74)(30 56 175 75)(91 220 137 192)(92 221 138 193)(93 222 139 194)(94 223 140 195)(95 224 141 196)(96 225 142 197)(97 226 143 198)(98 227 144 199)(99 228 145 200)(100 229 146 201)(101 230 147 202)(102 231 148 203)(103 232 149 204)(104 233 150 205)(105 234 121 206)(106 235 122 207)(107 236 123 208)(108 237 124 209)(109 238 125 210)(110 239 126 181)(111 240 127 182)(112 211 128 183)(113 212 129 184)(114 213 130 185)(115 214 131 186)(116 215 132 187)(117 216 133 188)(118 217 134 189)(119 218 135 190)(120 219 136 191)
(1 214 161 201)(2 233 162 190)(3 222 163 209)(4 211 164 198)(5 230 165 187)(6 219 166 206)(7 238 167 195)(8 227 168 184)(9 216 169 203)(10 235 170 192)(11 224 171 181)(12 213 172 200)(13 232 173 189)(14 221 174 208)(15 240 175 197)(16 229 176 186)(17 218 177 205)(18 237 178 194)(19 226 179 183)(20 215 180 202)(21 234 151 191)(22 223 152 210)(23 212 153 199)(24 231 154 188)(25 220 155 207)(26 239 156 196)(27 228 157 185)(28 217 158 204)(29 236 159 193)(30 225 160 182)(31 101 65 132)(32 120 66 121)(33 109 67 140)(34 98 68 129)(35 117 69 148)(36 106 70 137)(37 95 71 126)(38 114 72 145)(39 103 73 134)(40 92 74 123)(41 111 75 142)(42 100 76 131)(43 119 77 150)(44 108 78 139)(45 97 79 128)(46 116 80 147)(47 105 81 136)(48 94 82 125)(49 113 83 144)(50 102 84 133)(51 91 85 122)(52 110 86 141)(53 99 87 130)(54 118 88 149)(55 107 89 138)(56 96 90 127)(57 115 61 146)(58 104 62 135)(59 93 63 124)(60 112 64 143)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222222234444444444445566610···1012121212121215152020202020···2030···3060···60
size1111151515152226666101010103030222222···2442020202044444412···124···44···4

66 irreducible representations

dim11111112222222222444444444
type++++++++-++++++-++-++
imageC1C2C2C2C2C2C4S3D4Q8D5D6D6D10D10C4×S3C4×D5S3×D4S3×Q8S3×D5D4×D5Q8×D5D30.C2C2×S3×D5D15⋊Q8C20⋊D6
kernelD30.2Q8Dic155C4C3×C4⋊Dic5C5×C4⋊Dic3C2×D30.C2C2×C4×D15C4×D15C4⋊Dic5D30D30C4⋊Dic3C2×Dic5C2×C20C2×Dic3C2×C12C20C12C10C10C2×C4C6C6C4C22C2C2
# reps12112181222214248112224244

Matrix representation of D30.2Q8 in GL6(𝔽61)

60600000
19180000
0006000
001100
000010
000001
,
0170000
1800000
0006000
0060000
000010
000001
,
100000
010000
001000
000100
000001
0000600
,
43440000
19180000
0050000
0005000
0000219
00001959

G:=sub<GL(6,GF(61))| [60,19,0,0,0,0,60,18,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,18,0,0,0,0,17,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[43,19,0,0,0,0,44,18,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,2,19,0,0,0,0,19,59] >;

D30.2Q8 in GAP, Magma, Sage, TeX

D_{30}._2Q_8
% in TeX

G:=Group("D30.2Q8");
// GroupNames label

G:=SmallGroup(480,513);
// by ID

G=gap.SmallGroup(480,513);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^19,b*c=c*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽