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