metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.4Q8, C30.32D4, Dic15⋊3C4, C10.4Dic6, C6.4Dic10, C2.1Dic30, C22.4D30, C15⋊7(C4⋊C4), C6.8(C4×D5), (C2×C20).2S3, (C2×C60).3C2, C2.4(C4×D15), (C2×C4).1D15, (C2×C12).2D5, C10.15(C4×S3), C30.38(C2×C4), C5⋊4(Dic3⋊C4), (C2×C10).22D6, (C2×C6).22D10, C6.14(C5⋊D4), C3⋊3(C10.D4), C2.1(C15⋊7D4), C10.14(C3⋊D4), (C2×C30).23C22, (C2×Dic15).1C2, SmallGroup(240,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.4Q8
G = < a,b,c | a30=b4=1, c2=a15b2, ab=ba, cac-1=a-1, cbc-1=a15b-1 >
Smallest permutation representation of C30.4Q8
►Regular action 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)
(1 116 173 192)(2 117 174 193)(3 118 175 194)(4 119 176 195)(5 120 177 196)(6 91 178 197)(7 92 179 198)(8 93 180 199)(9 94 151 200)(10 95 152 201)(11 96 153 202)(12 97 154 203)(13 98 155 204)(14 99 156 205)(15 100 157 206)(16 101 158 207)(17 102 159 208)(18 103 160 209)(19 104 161 210)(20 105 162 181)(21 106 163 182)(22 107 164 183)(23 108 165 184)(24 109 166 185)(25 110 167 186)(26 111 168 187)(27 112 169 188)(28 113 170 189)(29 114 171 190)(30 115 172 191)(31 229 83 142)(32 230 84 143)(33 231 85 144)(34 232 86 145)(35 233 87 146)(36 234 88 147)(37 235 89 148)(38 236 90 149)(39 237 61 150)(40 238 62 121)(41 239 63 122)(42 240 64 123)(43 211 65 124)(44 212 66 125)(45 213 67 126)(46 214 68 127)(47 215 69 128)(48 216 70 129)(49 217 71 130)(50 218 72 131)(51 219 73 132)(52 220 74 133)(53 221 75 134)(54 222 76 135)(55 223 77 136)(56 224 78 137)(57 225 79 138)(58 226 80 139)(59 227 81 140)(60 228 82 141)
(1 51 158 88)(2 50 159 87)(3 49 160 86)(4 48 161 85)(5 47 162 84)(6 46 163 83)(7 45 164 82)(8 44 165 81)(9 43 166 80)(10 42 167 79)(11 41 168 78)(12 40 169 77)(13 39 170 76)(14 38 171 75)(15 37 172 74)(16 36 173 73)(17 35 174 72)(18 34 175 71)(19 33 176 70)(20 32 177 69)(21 31 178 68)(22 60 179 67)(23 59 180 66)(24 58 151 65)(25 57 152 64)(26 56 153 63)(27 55 154 62)(28 54 155 61)(29 53 156 90)(30 52 157 89)(91 142 182 214)(92 141 183 213)(93 140 184 212)(94 139 185 211)(95 138 186 240)(96 137 187 239)(97 136 188 238)(98 135 189 237)(99 134 190 236)(100 133 191 235)(101 132 192 234)(102 131 193 233)(103 130 194 232)(104 129 195 231)(105 128 196 230)(106 127 197 229)(107 126 198 228)(108 125 199 227)(109 124 200 226)(110 123 201 225)(111 122 202 224)(112 121 203 223)(113 150 204 222)(114 149 205 221)(115 148 206 220)(116 147 207 219)(117 146 208 218)(118 145 209 217)(119 144 210 216)(120 143 181 215)
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), (1,116,173,192)(2,117,174,193)(3,118,175,194)(4,119,176,195)(5,120,177,196)(6,91,178,197)(7,92,179,198)(8,93,180,199)(9,94,151,200)(10,95,152,201)(11,96,153,202)(12,97,154,203)(13,98,155,204)(14,99,156,205)(15,100,157,206)(16,101,158,207)(17,102,159,208)(18,103,160,209)(19,104,161,210)(20,105,162,181)(21,106,163,182)(22,107,164,183)(23,108,165,184)(24,109,166,185)(25,110,167,186)(26,111,168,187)(27,112,169,188)(28,113,170,189)(29,114,171,190)(30,115,172,191)(31,229,83,142)(32,230,84,143)(33,231,85,144)(34,232,86,145)(35,233,87,146)(36,234,88,147)(37,235,89,148)(38,236,90,149)(39,237,61,150)(40,238,62,121)(41,239,63,122)(42,240,64,123)(43,211,65,124)(44,212,66,125)(45,213,67,126)(46,214,68,127)(47,215,69,128)(48,216,70,129)(49,217,71,130)(50,218,72,131)(51,219,73,132)(52,220,74,133)(53,221,75,134)(54,222,76,135)(55,223,77,136)(56,224,78,137)(57,225,79,138)(58,226,80,139)(59,227,81,140)(60,228,82,141), (1,51,158,88)(2,50,159,87)(3,49,160,86)(4,48,161,85)(5,47,162,84)(6,46,163,83)(7,45,164,82)(8,44,165,81)(9,43,166,80)(10,42,167,79)(11,41,168,78)(12,40,169,77)(13,39,170,76)(14,38,171,75)(15,37,172,74)(16,36,173,73)(17,35,174,72)(18,34,175,71)(19,33,176,70)(20,32,177,69)(21,31,178,68)(22,60,179,67)(23,59,180,66)(24,58,151,65)(25,57,152,64)(26,56,153,63)(27,55,154,62)(28,54,155,61)(29,53,156,90)(30,52,157,89)(91,142,182,214)(92,141,183,213)(93,140,184,212)(94,139,185,211)(95,138,186,240)(96,137,187,239)(97,136,188,238)(98,135,189,237)(99,134,190,236)(100,133,191,235)(101,132,192,234)(102,131,193,233)(103,130,194,232)(104,129,195,231)(105,128,196,230)(106,127,197,229)(107,126,198,228)(108,125,199,227)(109,124,200,226)(110,123,201,225)(111,122,202,224)(112,121,203,223)(113,150,204,222)(114,149,205,221)(115,148,206,220)(116,147,207,219)(117,146,208,218)(118,145,209,217)(119,144,210,216)(120,143,181,215)>;
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), (1,116,173,192)(2,117,174,193)(3,118,175,194)(4,119,176,195)(5,120,177,196)(6,91,178,197)(7,92,179,198)(8,93,180,199)(9,94,151,200)(10,95,152,201)(11,96,153,202)(12,97,154,203)(13,98,155,204)(14,99,156,205)(15,100,157,206)(16,101,158,207)(17,102,159,208)(18,103,160,209)(19,104,161,210)(20,105,162,181)(21,106,163,182)(22,107,164,183)(23,108,165,184)(24,109,166,185)(25,110,167,186)(26,111,168,187)(27,112,169,188)(28,113,170,189)(29,114,171,190)(30,115,172,191)(31,229,83,142)(32,230,84,143)(33,231,85,144)(34,232,86,145)(35,233,87,146)(36,234,88,147)(37,235,89,148)(38,236,90,149)(39,237,61,150)(40,238,62,121)(41,239,63,122)(42,240,64,123)(43,211,65,124)(44,212,66,125)(45,213,67,126)(46,214,68,127)(47,215,69,128)(48,216,70,129)(49,217,71,130)(50,218,72,131)(51,219,73,132)(52,220,74,133)(53,221,75,134)(54,222,76,135)(55,223,77,136)(56,224,78,137)(57,225,79,138)(58,226,80,139)(59,227,81,140)(60,228,82,141), (1,51,158,88)(2,50,159,87)(3,49,160,86)(4,48,161,85)(5,47,162,84)(6,46,163,83)(7,45,164,82)(8,44,165,81)(9,43,166,80)(10,42,167,79)(11,41,168,78)(12,40,169,77)(13,39,170,76)(14,38,171,75)(15,37,172,74)(16,36,173,73)(17,35,174,72)(18,34,175,71)(19,33,176,70)(20,32,177,69)(21,31,178,68)(22,60,179,67)(23,59,180,66)(24,58,151,65)(25,57,152,64)(26,56,153,63)(27,55,154,62)(28,54,155,61)(29,53,156,90)(30,52,157,89)(91,142,182,214)(92,141,183,213)(93,140,184,212)(94,139,185,211)(95,138,186,240)(96,137,187,239)(97,136,188,238)(98,135,189,237)(99,134,190,236)(100,133,191,235)(101,132,192,234)(102,131,193,233)(103,130,194,232)(104,129,195,231)(105,128,196,230)(106,127,197,229)(107,126,198,228)(108,125,199,227)(109,124,200,226)(110,123,201,225)(111,122,202,224)(112,121,203,223)(113,150,204,222)(114,149,205,221)(115,148,206,220)(116,147,207,219)(117,146,208,218)(118,145,209,217)(119,144,210,216)(120,143,181,215) );
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)], [(1,116,173,192),(2,117,174,193),(3,118,175,194),(4,119,176,195),(5,120,177,196),(6,91,178,197),(7,92,179,198),(8,93,180,199),(9,94,151,200),(10,95,152,201),(11,96,153,202),(12,97,154,203),(13,98,155,204),(14,99,156,205),(15,100,157,206),(16,101,158,207),(17,102,159,208),(18,103,160,209),(19,104,161,210),(20,105,162,181),(21,106,163,182),(22,107,164,183),(23,108,165,184),(24,109,166,185),(25,110,167,186),(26,111,168,187),(27,112,169,188),(28,113,170,189),(29,114,171,190),(30,115,172,191),(31,229,83,142),(32,230,84,143),(33,231,85,144),(34,232,86,145),(35,233,87,146),(36,234,88,147),(37,235,89,148),(38,236,90,149),(39,237,61,150),(40,238,62,121),(41,239,63,122),(42,240,64,123),(43,211,65,124),(44,212,66,125),(45,213,67,126),(46,214,68,127),(47,215,69,128),(48,216,70,129),(49,217,71,130),(50,218,72,131),(51,219,73,132),(52,220,74,133),(53,221,75,134),(54,222,76,135),(55,223,77,136),(56,224,78,137),(57,225,79,138),(58,226,80,139),(59,227,81,140),(60,228,82,141)], [(1,51,158,88),(2,50,159,87),(3,49,160,86),(4,48,161,85),(5,47,162,84),(6,46,163,83),(7,45,164,82),(8,44,165,81),(9,43,166,80),(10,42,167,79),(11,41,168,78),(12,40,169,77),(13,39,170,76),(14,38,171,75),(15,37,172,74),(16,36,173,73),(17,35,174,72),(18,34,175,71),(19,33,176,70),(20,32,177,69),(21,31,178,68),(22,60,179,67),(23,59,180,66),(24,58,151,65),(25,57,152,64),(26,56,153,63),(27,55,154,62),(28,54,155,61),(29,53,156,90),(30,52,157,89),(91,142,182,214),(92,141,183,213),(93,140,184,212),(94,139,185,211),(95,138,186,240),(96,137,187,239),(97,136,188,238),(98,135,189,237),(99,134,190,236),(100,133,191,235),(101,132,192,234),(102,131,193,233),(103,130,194,232),(104,129,195,231),(105,128,196,230),(106,127,197,229),(107,126,198,228),(108,125,199,227),(109,124,200,226),(110,123,201,225),(111,122,202,224),(112,121,203,223),(113,150,204,222),(114,149,205,221),(115,148,206,220),(116,147,207,219),(117,146,208,218),(118,145,209,217),(119,144,210,216),(120,143,181,215)])
C30.4Q8 is a maximal subgroup of
Dic5⋊5Dic6 Dic3⋊5Dic10 Dic15⋊1Q8 Dic15⋊Q8 Dic5.2Dic6 Dic15.Q8 Dic15.2Q8 D6⋊C4.D5 Dic3.2Dic10 D10⋊Dic6 D6⋊Dic10 (C2×C12).D10 (C4×Dic3)⋊D5 C5⋊(C42⋊3S3) Dic5.7Dic6 Dic3.3Dic10 D5×Dic3⋊C4 (D5×Dic3)⋊C4 D6.(C4×D5) S3×C10.D4 (C6×D5).D4 Dic15⋊D4 D6⋊1Dic10 D10⋊1Dic6 (C2×D12).D5 Dic15.D4 D6⋊4Dic10 D6⋊(C4×D5) C15⋊17(C4×D4) D6⋊C4⋊D5 D10⋊C4⋊S3 Dic15⋊2D4 C4×Dic30 C60.24Q8 C42⋊2D15 C42⋊3D15 C23.15D30 C22⋊2Dic30 C23.8D30 Dic15⋊19D4 D30.28D4 D30⋊9D4 Dic15⋊10Q8 C4⋊Dic30 Dic15.3Q8 C4.Dic30 C4⋊C4×D15 D30.29D4 D30⋊5Q8 C4⋊C4⋊D15 C60.205D4 C4×C15⋊7D4 C23.28D30 C23.22D30 Dic15⋊12D4 Dic15⋊4Q8 D30⋊7Q8
C30.4Q8 is a maximal quotient of
C60.1Q8 C60.2Q8 C60.26Q8 C60.210D4 C30.29C42
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | D5 | D6 | D10 | Dic6 | C4×S3 | C3⋊D4 | D15 | Dic10 | C4×D5 | C5⋊D4 | D30 | Dic30 | C4×D15 | C15⋊7D4 |
| kernel | C30.4Q8 | C2×Dic15 | C2×C60 | Dic15 | C2×C20 | C30 | C30 | C2×C12 | C2×C10 | C2×C6 | C10 | C10 | C10 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C30.4Q8 ►in GL4(𝔽61) generated by
| 1 | 60 | 0 | 0 |
| 19 | 43 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 1 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 9 | 43 |
| 0 | 0 | 18 | 52 |
| 0 | 44 | 0 | 0 |
| 43 | 0 | 0 | 0 |
| 0 | 0 | 52 | 52 |
| 0 | 0 | 43 | 9 |
G:=sub<GL(4,GF(61))| [1,19,0,0,60,43,0,0,0,0,0,60,0,0,1,1],[50,0,0,0,0,50,0,0,0,0,9,18,0,0,43,52],[0,43,0,0,44,0,0,0,0,0,52,43,0,0,52,9] >;
C30.4Q8 in GAP, Magma, Sage, TeX
C_{30}._4Q_8 % in TeX
G:=Group("C30.4Q8"); // GroupNames label
G:=SmallGroup(240,73);
// by ID
G=gap.SmallGroup(240,73);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,121,31,964,6917]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=1,c^2=a^15*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^15*b^-1>;
// generators/relations
Export