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G = D30.Q8order 480 = 25·3·5

1st non-split extension by D30 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.1Q8, D30.36D4, C6.57(D4×D5), D152(C4⋊C4), C6.12(Q8×D5), D30.C23C4, Dic36(C4×D5), Dic58(C4×S3), C10.59(S3×D4), C10.12(S3×Q8), C30.36(C2×Q8), Dic3⋊C414D5, D30.31(C2×C4), C30.127(C2×D4), (C2×C20).195D6, C2.4(D15⋊Q8), C10.D413S3, (C2×C12).194D10, (C2×C30).94C23, C30.55(C22×C4), C30.Q814C2, C6.Dic1015C2, C2.2(D10⋊D6), (C2×C60).166C22, (C2×Dic3).95D10, (C2×Dic5).104D6, (C6×Dic5).55C22, (C10×Dic3).54C22, (C2×Dic15).204C22, (C22×D15).103C22, C52(S3×C4⋊C4), C31(D5×C4⋊C4), C157(C2×C4⋊C4), C6.23(C2×C4×D5), C2.25(C4×S3×D5), C10.56(S3×C2×C4), (C2×C4×D15).8C2, C22.49(C2×S3×D5), (C5×Dic3)⋊9(C2×C4), (C3×Dic5)⋊2(C2×C4), (C2×C4).179(S3×D5), (C5×Dic3⋊C4)⋊14C2, (C2×D30.C2).3C2, (C3×C10.D4)⋊13C2, (C2×C6).106(C22×D5), (C2×C10).106(C22×S3), SmallGroup(480,480)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D30.Q8
 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=a15c-1 >

Subgroups: 940 in 184 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×13], C23, D5 [×4], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×4], D6 [×6], C2×C6, C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×4], 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, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4 [×3], C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15, C60, D30 [×6], C2×C30, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C4×D5 [×3], S3×C4⋊C4, D30.C2 [×4], D30.C2 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C4×D15 [×2], C2×Dic15, C2×C60, C22×D15, D5×C4⋊C4, C30.Q8, C6.Dic10, C3×C10.D4, C5×Dic3⋊C4, C2×D30.C2 [×2], C2×C4×D15, D30.Q8
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, C2×S3×D5, D5×C4⋊C4, D15⋊Q8, C4×S3×D5, D10⋊D6, D30.Q8

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

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

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

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

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

dim111111112222222222444444444
type+++++++++-++++++-++-++
imageC1C2C2C2C2C2C2C4S3D4Q8D5D6D6D10D10C4×S3C4×D5S3×D4S3×Q8S3×D5D4×D5Q8×D5C2×S3×D5D15⋊Q8C4×S3×D5D10⋊D6
kernelD30.Q8C30.Q8C6.Dic10C3×C10.D4C5×Dic3⋊C4C2×D30.C2C2×C4×D15D30.C2C10.D4D30D30Dic3⋊C4C2×Dic5C2×C20C2×Dic3C2×C12Dic5Dic3C10C10C2×C4C6C6C22C2C2C2
# reps111112181222214248112222444

Matrix representation of D30.Q8 in GL6(𝔽61)

1600000
100000
00176000
001000
000010
000001
,
0600000
6000000
00472400
00301400
000010
000001
,
5000000
0500000
001000
000100
000013
00004060
,
6000000
0600000
00372800
00472400
00001757
00004244

G:=sub<GL(6,GF(61))| [1,1,0,0,0,0,60,0,0,0,0,0,0,0,17,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,47,30,0,0,0,0,24,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,3,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,37,47,0,0,0,0,28,24,0,0,0,0,0,0,17,42,0,0,0,0,57,44] >;

D30.Q8 in GAP, Magma, Sage, TeX

D_{30}.Q_8
% in TeX

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

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

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

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,422,219,58,1356,18822]);
// 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=a^15*c^-1>;
// generators/relations

׿
×
𝔽