metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.1Q8, D30.36D4, C6.57(D4×D5), D15⋊2(C4⋊C4), C6.12(Q8×D5), D30.C2⋊3C4, Dic3⋊6(C4×D5), Dic5⋊8(C4×S3), C10.59(S3×D4), C10.12(S3×Q8), C30.36(C2×Q8), Dic3⋊C4⋊14D5, D30.31(C2×C4), C30.127(C2×D4), (C2×C20).195D6, C2.4(D15⋊Q8), C10.D4⋊13S3, (C2×C12).194D10, (C2×C30).94C23, C30.55(C22×C4), C30.Q8⋊14C2, C6.Dic10⋊15C2, 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, C5⋊2(S3×C4⋊C4), C3⋊1(D5×C4⋊C4), C15⋊7(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
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, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, Dic5, 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, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, D30, C2×C30, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C4×D5, S3×C4⋊C4, D30.C2, D30.C2, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×C4⋊C4, C30.Q8, C6.Dic10, C3×C10.D4, C5×Dic3⋊C4, C2×D30.C2, C2×C4×D15, D30.Q8
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, C2×S3×D5, D5×C4⋊C4, D15⋊Q8, C4×S3×D5, D10⋊D6, 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 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 90)(82 89)(83 88)(84 87)(85 86)(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)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)(141 144)(142 143)(151 174)(152 173)(153 172)(154 171)(155 170)(156 169)(157 168)(158 167)(159 166)(160 165)(161 164)(162 163)(175 180)(176 179)(177 178)(181 184)(182 183)(185 210)(186 209)(187 208)(188 207)(189 206)(190 205)(191 204)(192 203)(193 202)(194 201)(195 200)(196 199)(197 198)(211 232)(212 231)(213 230)(214 229)(215 228)(216 227)(217 226)(218 225)(219 224)(220 223)(221 222)(233 240)(234 239)(235 238)(236 237)
(1 71 222 198)(2 72 223 199)(3 73 224 200)(4 74 225 201)(5 75 226 202)(6 76 227 203)(7 77 228 204)(8 78 229 205)(9 79 230 206)(10 80 231 207)(11 81 232 208)(12 82 233 209)(13 83 234 210)(14 84 235 181)(15 85 236 182)(16 86 237 183)(17 87 238 184)(18 88 239 185)(19 89 240 186)(20 90 211 187)(21 61 212 188)(22 62 213 189)(23 63 214 190)(24 64 215 191)(25 65 216 192)(26 66 217 193)(27 67 218 194)(28 68 219 195)(29 69 220 196)(30 70 221 197)(31 115 169 134)(32 116 170 135)(33 117 171 136)(34 118 172 137)(35 119 173 138)(36 120 174 139)(37 91 175 140)(38 92 176 141)(39 93 177 142)(40 94 178 143)(41 95 179 144)(42 96 180 145)(43 97 151 146)(44 98 152 147)(45 99 153 148)(46 100 154 149)(47 101 155 150)(48 102 156 121)(49 103 157 122)(50 104 158 123)(51 105 159 124)(52 106 160 125)(53 107 161 126)(54 108 162 127)(55 109 163 128)(56 110 164 129)(57 111 165 130)(58 112 166 131)(59 113 167 132)(60 114 168 133)
(1 109 237 143)(2 98 238 132)(3 117 239 121)(4 106 240 140)(5 95 211 129)(6 114 212 148)(7 103 213 137)(8 92 214 126)(9 111 215 145)(10 100 216 134)(11 119 217 123)(12 108 218 142)(13 97 219 131)(14 116 220 150)(15 105 221 139)(16 94 222 128)(17 113 223 147)(18 102 224 136)(19 91 225 125)(20 110 226 144)(21 99 227 133)(22 118 228 122)(23 107 229 141)(24 96 230 130)(25 115 231 149)(26 104 232 138)(27 93 233 127)(28 112 234 146)(29 101 235 135)(30 120 236 124)(31 192 154 80)(32 181 155 69)(33 200 156 88)(34 189 157 77)(35 208 158 66)(36 197 159 85)(37 186 160 74)(38 205 161 63)(39 194 162 82)(40 183 163 71)(41 202 164 90)(42 191 165 79)(43 210 166 68)(44 199 167 87)(45 188 168 76)(46 207 169 65)(47 196 170 84)(48 185 171 73)(49 204 172 62)(50 193 173 81)(51 182 174 70)(52 201 175 89)(53 190 176 78)(54 209 177 67)(55 198 178 86)(56 187 179 75)(57 206 180 64)(58 195 151 83)(59 184 152 72)(60 203 153 61)
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)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,90)(82,89)(83,88)(84,87)(85,86)(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)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(151,174)(152,173)(153,172)(154,171)(155,170)(156,169)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163)(175,180)(176,179)(177,178)(181,184)(182,183)(185,210)(186,209)(187,208)(188,207)(189,206)(190,205)(191,204)(192,203)(193,202)(194,201)(195,200)(196,199)(197,198)(211,232)(212,231)(213,230)(214,229)(215,228)(216,227)(217,226)(218,225)(219,224)(220,223)(221,222)(233,240)(234,239)(235,238)(236,237), (1,71,222,198)(2,72,223,199)(3,73,224,200)(4,74,225,201)(5,75,226,202)(6,76,227,203)(7,77,228,204)(8,78,229,205)(9,79,230,206)(10,80,231,207)(11,81,232,208)(12,82,233,209)(13,83,234,210)(14,84,235,181)(15,85,236,182)(16,86,237,183)(17,87,238,184)(18,88,239,185)(19,89,240,186)(20,90,211,187)(21,61,212,188)(22,62,213,189)(23,63,214,190)(24,64,215,191)(25,65,216,192)(26,66,217,193)(27,67,218,194)(28,68,219,195)(29,69,220,196)(30,70,221,197)(31,115,169,134)(32,116,170,135)(33,117,171,136)(34,118,172,137)(35,119,173,138)(36,120,174,139)(37,91,175,140)(38,92,176,141)(39,93,177,142)(40,94,178,143)(41,95,179,144)(42,96,180,145)(43,97,151,146)(44,98,152,147)(45,99,153,148)(46,100,154,149)(47,101,155,150)(48,102,156,121)(49,103,157,122)(50,104,158,123)(51,105,159,124)(52,106,160,125)(53,107,161,126)(54,108,162,127)(55,109,163,128)(56,110,164,129)(57,111,165,130)(58,112,166,131)(59,113,167,132)(60,114,168,133), (1,109,237,143)(2,98,238,132)(3,117,239,121)(4,106,240,140)(5,95,211,129)(6,114,212,148)(7,103,213,137)(8,92,214,126)(9,111,215,145)(10,100,216,134)(11,119,217,123)(12,108,218,142)(13,97,219,131)(14,116,220,150)(15,105,221,139)(16,94,222,128)(17,113,223,147)(18,102,224,136)(19,91,225,125)(20,110,226,144)(21,99,227,133)(22,118,228,122)(23,107,229,141)(24,96,230,130)(25,115,231,149)(26,104,232,138)(27,93,233,127)(28,112,234,146)(29,101,235,135)(30,120,236,124)(31,192,154,80)(32,181,155,69)(33,200,156,88)(34,189,157,77)(35,208,158,66)(36,197,159,85)(37,186,160,74)(38,205,161,63)(39,194,162,82)(40,183,163,71)(41,202,164,90)(42,191,165,79)(43,210,166,68)(44,199,167,87)(45,188,168,76)(46,207,169,65)(47,196,170,84)(48,185,171,73)(49,204,172,62)(50,193,173,81)(51,182,174,70)(52,201,175,89)(53,190,176,78)(54,209,177,67)(55,198,178,86)(56,187,179,75)(57,206,180,64)(58,195,151,83)(59,184,152,72)(60,203,153,61)>;
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)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,90)(82,89)(83,88)(84,87)(85,86)(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)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(151,174)(152,173)(153,172)(154,171)(155,170)(156,169)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163)(175,180)(176,179)(177,178)(181,184)(182,183)(185,210)(186,209)(187,208)(188,207)(189,206)(190,205)(191,204)(192,203)(193,202)(194,201)(195,200)(196,199)(197,198)(211,232)(212,231)(213,230)(214,229)(215,228)(216,227)(217,226)(218,225)(219,224)(220,223)(221,222)(233,240)(234,239)(235,238)(236,237), (1,71,222,198)(2,72,223,199)(3,73,224,200)(4,74,225,201)(5,75,226,202)(6,76,227,203)(7,77,228,204)(8,78,229,205)(9,79,230,206)(10,80,231,207)(11,81,232,208)(12,82,233,209)(13,83,234,210)(14,84,235,181)(15,85,236,182)(16,86,237,183)(17,87,238,184)(18,88,239,185)(19,89,240,186)(20,90,211,187)(21,61,212,188)(22,62,213,189)(23,63,214,190)(24,64,215,191)(25,65,216,192)(26,66,217,193)(27,67,218,194)(28,68,219,195)(29,69,220,196)(30,70,221,197)(31,115,169,134)(32,116,170,135)(33,117,171,136)(34,118,172,137)(35,119,173,138)(36,120,174,139)(37,91,175,140)(38,92,176,141)(39,93,177,142)(40,94,178,143)(41,95,179,144)(42,96,180,145)(43,97,151,146)(44,98,152,147)(45,99,153,148)(46,100,154,149)(47,101,155,150)(48,102,156,121)(49,103,157,122)(50,104,158,123)(51,105,159,124)(52,106,160,125)(53,107,161,126)(54,108,162,127)(55,109,163,128)(56,110,164,129)(57,111,165,130)(58,112,166,131)(59,113,167,132)(60,114,168,133), (1,109,237,143)(2,98,238,132)(3,117,239,121)(4,106,240,140)(5,95,211,129)(6,114,212,148)(7,103,213,137)(8,92,214,126)(9,111,215,145)(10,100,216,134)(11,119,217,123)(12,108,218,142)(13,97,219,131)(14,116,220,150)(15,105,221,139)(16,94,222,128)(17,113,223,147)(18,102,224,136)(19,91,225,125)(20,110,226,144)(21,99,227,133)(22,118,228,122)(23,107,229,141)(24,96,230,130)(25,115,231,149)(26,104,232,138)(27,93,233,127)(28,112,234,146)(29,101,235,135)(30,120,236,124)(31,192,154,80)(32,181,155,69)(33,200,156,88)(34,189,157,77)(35,208,158,66)(36,197,159,85)(37,186,160,74)(38,205,161,63)(39,194,162,82)(40,183,163,71)(41,202,164,90)(42,191,165,79)(43,210,166,68)(44,199,167,87)(45,188,168,76)(46,207,169,65)(47,196,170,84)(48,185,171,73)(49,204,172,62)(50,193,173,81)(51,182,174,70)(52,201,175,89)(53,190,176,78)(54,209,177,67)(55,198,178,86)(56,187,179,75)(57,206,180,64)(58,195,151,83)(59,184,152,72)(60,203,153,61) );
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),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,90),(82,89),(83,88),(84,87),(85,86),(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),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(135,150),(136,149),(137,148),(138,147),(139,146),(140,145),(141,144),(142,143),(151,174),(152,173),(153,172),(154,171),(155,170),(156,169),(157,168),(158,167),(159,166),(160,165),(161,164),(162,163),(175,180),(176,179),(177,178),(181,184),(182,183),(185,210),(186,209),(187,208),(188,207),(189,206),(190,205),(191,204),(192,203),(193,202),(194,201),(195,200),(196,199),(197,198),(211,232),(212,231),(213,230),(214,229),(215,228),(216,227),(217,226),(218,225),(219,224),(220,223),(221,222),(233,240),(234,239),(235,238),(236,237)], [(1,71,222,198),(2,72,223,199),(3,73,224,200),(4,74,225,201),(5,75,226,202),(6,76,227,203),(7,77,228,204),(8,78,229,205),(9,79,230,206),(10,80,231,207),(11,81,232,208),(12,82,233,209),(13,83,234,210),(14,84,235,181),(15,85,236,182),(16,86,237,183),(17,87,238,184),(18,88,239,185),(19,89,240,186),(20,90,211,187),(21,61,212,188),(22,62,213,189),(23,63,214,190),(24,64,215,191),(25,65,216,192),(26,66,217,193),(27,67,218,194),(28,68,219,195),(29,69,220,196),(30,70,221,197),(31,115,169,134),(32,116,170,135),(33,117,171,136),(34,118,172,137),(35,119,173,138),(36,120,174,139),(37,91,175,140),(38,92,176,141),(39,93,177,142),(40,94,178,143),(41,95,179,144),(42,96,180,145),(43,97,151,146),(44,98,152,147),(45,99,153,148),(46,100,154,149),(47,101,155,150),(48,102,156,121),(49,103,157,122),(50,104,158,123),(51,105,159,124),(52,106,160,125),(53,107,161,126),(54,108,162,127),(55,109,163,128),(56,110,164,129),(57,111,165,130),(58,112,166,131),(59,113,167,132),(60,114,168,133)], [(1,109,237,143),(2,98,238,132),(3,117,239,121),(4,106,240,140),(5,95,211,129),(6,114,212,148),(7,103,213,137),(8,92,214,126),(9,111,215,145),(10,100,216,134),(11,119,217,123),(12,108,218,142),(13,97,219,131),(14,116,220,150),(15,105,221,139),(16,94,222,128),(17,113,223,147),(18,102,224,136),(19,91,225,125),(20,110,226,144),(21,99,227,133),(22,118,228,122),(23,107,229,141),(24,96,230,130),(25,115,231,149),(26,104,232,138),(27,93,233,127),(28,112,234,146),(29,101,235,135),(30,120,236,124),(31,192,154,80),(32,181,155,69),(33,200,156,88),(34,189,157,77),(35,208,158,66),(36,197,159,85),(37,186,160,74),(38,205,161,63),(39,194,162,82),(40,183,163,71),(41,202,164,90),(42,191,165,79),(43,210,166,68),(44,199,167,87),(45,188,168,76),(46,207,169,65),(47,196,170,84),(48,185,171,73),(49,204,172,62),(50,193,173,81),(51,182,174,70),(52,201,175,89),(53,190,176,78),(54,209,177,67),(55,198,178,86),(56,187,179,75),(57,206,180,64),(58,195,151,83),(59,184,152,72),(60,203,153,61)]])
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 | 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 | C2 | C4 | S3 | D4 | Q8 | D5 | D6 | D6 | D10 | D10 | C4×S3 | C4×D5 | S3×D4 | S3×Q8 | S3×D5 | D4×D5 | Q8×D5 | C2×S3×D5 | D15⋊Q8 | C4×S3×D5 | D10⋊D6 |
| kernel | D30.Q8 | C30.Q8 | C6.Dic10 | C3×C10.D4 | C5×Dic3⋊C4 | C2×D30.C2 | C2×C4×D15 | D30.C2 | C10.D4 | D30 | D30 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | Dic5 | Dic3 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30.Q8 ►in GL6(𝔽61)
| 1 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 1 | 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 | 24 | 0 | 0 |
| 0 | 0 | 30 | 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 | 3 |
| 0 | 0 | 0 | 0 | 40 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 28 | 0 | 0 |
| 0 | 0 | 47 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 57 |
| 0 | 0 | 0 | 0 | 42 | 44 |
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