metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.2Q8, D30.26D4, C20⋊8(C4×S3), C12⋊4(C4×D5), C60⋊22(C2×C4), (C4×D15)⋊9C4, C6.43(D4×D5), D15⋊3(C4⋊C4), C6.17(Q8×D5), C4⋊Dic5⋊14S3, C4⋊Dic3⋊14D5, C10.44(S3×D4), C30.55(C2×D4), C10.17(S3×Q8), C30.51(C2×Q8), C4⋊3(D30.C2), D30.41(C2×C4), (C2×C20).128D6, C2.5(D15⋊Q8), Dic15⋊22(C2×C4), (C2×C12).129D10, C2.3(C20⋊D6), Dic15⋊5C4⋊23C2, (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, C5⋊3(S3×C4⋊C4), C3⋊2(D5×C4⋊C4), C15⋊10(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
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, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C2×C4⋊C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, D30, C2×C30, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C4×D5, S3×C4⋊C4, D30.C2, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×C4⋊C4, Dic15⋊5C4, C3×C4⋊Dic5, C5×C4⋊Dic3, C2×D30.C2, C2×C4×D15, D30.2Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D5, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C4×S3, C22×S3, C2×C4⋊C4, C4×D5, C22×D5, S3×C2×C4, S3×D4, S3×Q8, S3×D5, C2×C4×D5, D4×D5, Q8×D5, S3×C4⋊C4, D30.C2, C2×S3×D5, D5×C4⋊C4, D15⋊Q8, C20⋊D6, C2×D30.C2, 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 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(59 60)(61 62)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(111 120)(112 119)(113 118)(114 117)(115 116)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)(151 180)(152 179)(153 178)(154 177)(155 176)(156 175)(157 174)(158 173)(159 172)(160 171)(161 170)(162 169)(163 168)(164 167)(165 166)(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)(211 228)(212 227)(213 226)(214 225)(215 224)(216 223)(217 222)(218 221)(219 220)(229 240)(230 239)(231 238)(232 237)(233 236)(234 235)
(1 116 184 77)(2 117 185 78)(3 118 186 79)(4 119 187 80)(5 120 188 81)(6 91 189 82)(7 92 190 83)(8 93 191 84)(9 94 192 85)(10 95 193 86)(11 96 194 87)(12 97 195 88)(13 98 196 89)(14 99 197 90)(15 100 198 61)(16 101 199 62)(17 102 200 63)(18 103 201 64)(19 104 202 65)(20 105 203 66)(21 106 204 67)(22 107 205 68)(23 108 206 69)(24 109 207 70)(25 110 208 71)(26 111 209 72)(27 112 210 73)(28 113 181 74)(29 114 182 75)(30 115 183 76)(31 128 236 152)(32 129 237 153)(33 130 238 154)(34 131 239 155)(35 132 240 156)(36 133 211 157)(37 134 212 158)(38 135 213 159)(39 136 214 160)(40 137 215 161)(41 138 216 162)(42 139 217 163)(43 140 218 164)(44 141 219 165)(45 142 220 166)(46 143 221 167)(47 144 222 168)(48 145 223 169)(49 146 224 170)(50 147 225 171)(51 148 226 172)(52 149 227 173)(53 150 228 174)(54 121 229 175)(55 122 230 176)(56 123 231 177)(57 124 232 178)(58 125 233 179)(59 126 234 180)(60 127 235 151)
(1 166 199 127)(2 155 200 146)(3 174 201 135)(4 163 202 124)(5 152 203 143)(6 171 204 132)(7 160 205 121)(8 179 206 140)(9 168 207 129)(10 157 208 148)(11 176 209 137)(12 165 210 126)(13 154 181 145)(14 173 182 134)(15 162 183 123)(16 151 184 142)(17 170 185 131)(18 159 186 150)(19 178 187 139)(20 167 188 128)(21 156 189 147)(22 175 190 136)(23 164 191 125)(24 153 192 144)(25 172 193 133)(26 161 194 122)(27 180 195 141)(28 169 196 130)(29 158 197 149)(30 177 198 138)(31 105 221 81)(32 94 222 70)(33 113 223 89)(34 102 224 78)(35 91 225 67)(36 110 226 86)(37 99 227 75)(38 118 228 64)(39 107 229 83)(40 96 230 72)(41 115 231 61)(42 104 232 80)(43 93 233 69)(44 112 234 88)(45 101 235 77)(46 120 236 66)(47 109 237 85)(48 98 238 74)(49 117 239 63)(50 106 240 82)(51 95 211 71)(52 114 212 90)(53 103 213 79)(54 92 214 68)(55 111 215 87)(56 100 216 76)(57 119 217 65)(58 108 218 84)(59 97 219 73)(60 116 220 62)
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)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(59,60)(61,62)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)(111,120)(112,119)(113,118)(114,117)(115,116)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166)(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)(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)(219,220)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235), (1,116,184,77)(2,117,185,78)(3,118,186,79)(4,119,187,80)(5,120,188,81)(6,91,189,82)(7,92,190,83)(8,93,191,84)(9,94,192,85)(10,95,193,86)(11,96,194,87)(12,97,195,88)(13,98,196,89)(14,99,197,90)(15,100,198,61)(16,101,199,62)(17,102,200,63)(18,103,201,64)(19,104,202,65)(20,105,203,66)(21,106,204,67)(22,107,205,68)(23,108,206,69)(24,109,207,70)(25,110,208,71)(26,111,209,72)(27,112,210,73)(28,113,181,74)(29,114,182,75)(30,115,183,76)(31,128,236,152)(32,129,237,153)(33,130,238,154)(34,131,239,155)(35,132,240,156)(36,133,211,157)(37,134,212,158)(38,135,213,159)(39,136,214,160)(40,137,215,161)(41,138,216,162)(42,139,217,163)(43,140,218,164)(44,141,219,165)(45,142,220,166)(46,143,221,167)(47,144,222,168)(48,145,223,169)(49,146,224,170)(50,147,225,171)(51,148,226,172)(52,149,227,173)(53,150,228,174)(54,121,229,175)(55,122,230,176)(56,123,231,177)(57,124,232,178)(58,125,233,179)(59,126,234,180)(60,127,235,151), (1,166,199,127)(2,155,200,146)(3,174,201,135)(4,163,202,124)(5,152,203,143)(6,171,204,132)(7,160,205,121)(8,179,206,140)(9,168,207,129)(10,157,208,148)(11,176,209,137)(12,165,210,126)(13,154,181,145)(14,173,182,134)(15,162,183,123)(16,151,184,142)(17,170,185,131)(18,159,186,150)(19,178,187,139)(20,167,188,128)(21,156,189,147)(22,175,190,136)(23,164,191,125)(24,153,192,144)(25,172,193,133)(26,161,194,122)(27,180,195,141)(28,169,196,130)(29,158,197,149)(30,177,198,138)(31,105,221,81)(32,94,222,70)(33,113,223,89)(34,102,224,78)(35,91,225,67)(36,110,226,86)(37,99,227,75)(38,118,228,64)(39,107,229,83)(40,96,230,72)(41,115,231,61)(42,104,232,80)(43,93,233,69)(44,112,234,88)(45,101,235,77)(46,120,236,66)(47,109,237,85)(48,98,238,74)(49,117,239,63)(50,106,240,82)(51,95,211,71)(52,114,212,90)(53,103,213,79)(54,92,214,68)(55,111,215,87)(56,100,216,76)(57,119,217,65)(58,108,218,84)(59,97,219,73)(60,116,220,62)>;
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)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(59,60)(61,62)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)(111,120)(112,119)(113,118)(114,117)(115,116)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166)(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)(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)(219,220)(229,240)(230,239)(231,238)(232,237)(233,236)(234,235), (1,116,184,77)(2,117,185,78)(3,118,186,79)(4,119,187,80)(5,120,188,81)(6,91,189,82)(7,92,190,83)(8,93,191,84)(9,94,192,85)(10,95,193,86)(11,96,194,87)(12,97,195,88)(13,98,196,89)(14,99,197,90)(15,100,198,61)(16,101,199,62)(17,102,200,63)(18,103,201,64)(19,104,202,65)(20,105,203,66)(21,106,204,67)(22,107,205,68)(23,108,206,69)(24,109,207,70)(25,110,208,71)(26,111,209,72)(27,112,210,73)(28,113,181,74)(29,114,182,75)(30,115,183,76)(31,128,236,152)(32,129,237,153)(33,130,238,154)(34,131,239,155)(35,132,240,156)(36,133,211,157)(37,134,212,158)(38,135,213,159)(39,136,214,160)(40,137,215,161)(41,138,216,162)(42,139,217,163)(43,140,218,164)(44,141,219,165)(45,142,220,166)(46,143,221,167)(47,144,222,168)(48,145,223,169)(49,146,224,170)(50,147,225,171)(51,148,226,172)(52,149,227,173)(53,150,228,174)(54,121,229,175)(55,122,230,176)(56,123,231,177)(57,124,232,178)(58,125,233,179)(59,126,234,180)(60,127,235,151), (1,166,199,127)(2,155,200,146)(3,174,201,135)(4,163,202,124)(5,152,203,143)(6,171,204,132)(7,160,205,121)(8,179,206,140)(9,168,207,129)(10,157,208,148)(11,176,209,137)(12,165,210,126)(13,154,181,145)(14,173,182,134)(15,162,183,123)(16,151,184,142)(17,170,185,131)(18,159,186,150)(19,178,187,139)(20,167,188,128)(21,156,189,147)(22,175,190,136)(23,164,191,125)(24,153,192,144)(25,172,193,133)(26,161,194,122)(27,180,195,141)(28,169,196,130)(29,158,197,149)(30,177,198,138)(31,105,221,81)(32,94,222,70)(33,113,223,89)(34,102,224,78)(35,91,225,67)(36,110,226,86)(37,99,227,75)(38,118,228,64)(39,107,229,83)(40,96,230,72)(41,115,231,61)(42,104,232,80)(43,93,233,69)(44,112,234,88)(45,101,235,77)(46,120,236,66)(47,109,237,85)(48,98,238,74)(49,117,239,63)(50,106,240,82)(51,95,211,71)(52,114,212,90)(53,103,213,79)(54,92,214,68)(55,111,215,87)(56,100,216,76)(57,119,217,65)(58,108,218,84)(59,97,219,73)(60,116,220,62) );
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),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,52),(38,51),(39,50),(40,49),(41,48),(42,47),(43,46),(44,45),(59,60),(61,62),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(91,110),(92,109),(93,108),(94,107),(95,106),(96,105),(97,104),(98,103),(99,102),(100,101),(111,120),(112,119),(113,118),(114,117),(115,116),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(133,150),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,143),(141,142),(151,180),(152,179),(153,178),(154,177),(155,176),(156,175),(157,174),(158,173),(159,172),(160,171),(161,170),(162,169),(163,168),(164,167),(165,166),(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),(211,228),(212,227),(213,226),(214,225),(215,224),(216,223),(217,222),(218,221),(219,220),(229,240),(230,239),(231,238),(232,237),(233,236),(234,235)], [(1,116,184,77),(2,117,185,78),(3,118,186,79),(4,119,187,80),(5,120,188,81),(6,91,189,82),(7,92,190,83),(8,93,191,84),(9,94,192,85),(10,95,193,86),(11,96,194,87),(12,97,195,88),(13,98,196,89),(14,99,197,90),(15,100,198,61),(16,101,199,62),(17,102,200,63),(18,103,201,64),(19,104,202,65),(20,105,203,66),(21,106,204,67),(22,107,205,68),(23,108,206,69),(24,109,207,70),(25,110,208,71),(26,111,209,72),(27,112,210,73),(28,113,181,74),(29,114,182,75),(30,115,183,76),(31,128,236,152),(32,129,237,153),(33,130,238,154),(34,131,239,155),(35,132,240,156),(36,133,211,157),(37,134,212,158),(38,135,213,159),(39,136,214,160),(40,137,215,161),(41,138,216,162),(42,139,217,163),(43,140,218,164),(44,141,219,165),(45,142,220,166),(46,143,221,167),(47,144,222,168),(48,145,223,169),(49,146,224,170),(50,147,225,171),(51,148,226,172),(52,149,227,173),(53,150,228,174),(54,121,229,175),(55,122,230,176),(56,123,231,177),(57,124,232,178),(58,125,233,179),(59,126,234,180),(60,127,235,151)], [(1,166,199,127),(2,155,200,146),(3,174,201,135),(4,163,202,124),(5,152,203,143),(6,171,204,132),(7,160,205,121),(8,179,206,140),(9,168,207,129),(10,157,208,148),(11,176,209,137),(12,165,210,126),(13,154,181,145),(14,173,182,134),(15,162,183,123),(16,151,184,142),(17,170,185,131),(18,159,186,150),(19,178,187,139),(20,167,188,128),(21,156,189,147),(22,175,190,136),(23,164,191,125),(24,153,192,144),(25,172,193,133),(26,161,194,122),(27,180,195,141),(28,169,196,130),(29,158,197,149),(30,177,198,138),(31,105,221,81),(32,94,222,70),(33,113,223,89),(34,102,224,78),(35,91,225,67),(36,110,226,86),(37,99,227,75),(38,118,228,64),(39,107,229,83),(40,96,230,72),(41,115,231,61),(42,104,232,80),(43,93,233,69),(44,112,234,88),(45,101,235,77),(46,120,236,66),(47,109,237,85),(48,98,238,74),(49,117,239,63),(50,106,240,82),(51,95,211,71),(52,114,212,90),(53,103,213,79),(54,92,214,68),(55,111,215,87),(56,100,216,76),(57,119,217,65),(58,108,218,84),(59,97,219,73),(60,116,220,62)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D5 | D6 | D6 | D10 | D10 | C4×S3 | C4×D5 | S3×D4 | S3×Q8 | S3×D5 | D4×D5 | Q8×D5 | D30.C2 | C2×S3×D5 | D15⋊Q8 | C20⋊D6 |
| kernel | D30.2Q8 | Dic15⋊5C4 | C3×C4⋊Dic5 | C5×C4⋊Dic3 | C2×D30.C2 | C2×C4×D15 | C4×D15 | C4⋊Dic5 | D30 | D30 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of D30.2Q8 ►in GL6(𝔽61)
| 60 | 60 | 0 | 0 | 0 | 0 |
| 19 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 18 | 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 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 43 | 44 | 0 | 0 | 0 | 0 |
| 19 | 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 |
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