metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.1Q8, C30.16D4, C10.12D12, C6.1Dic10, C10.1Dic6, Dic5⋊1Dic3, C15⋊4(C4⋊C4), C5⋊2(C4⋊Dic3), C6.14(C4×D5), (C2×C6).9D10, (C2×C10).9D6, C2.1(C15⋊Q8), C30.32(C2×C4), (C3×Dic5)⋊2C4, C6.6(C5⋊D4), C2.5(D5×Dic3), C22.8(S3×D5), C2.3(C5⋊D12), C3⋊1(C10.D4), (C2×C30).6C22, (C6×Dic5).2C2, (C2×Dic3).1D5, (C2×Dic5).1S3, (C2×Dic15).5C2, (C10×Dic3).2C2, C10.12(C2×Dic3), SmallGroup(240,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C30.Q8
G = < a,b,c | a30=b4=1, c2=a15b2, bab-1=a11, cac-1=a19, cbc-1=a15b-1 >
Smallest permutation representation of C30.Q8
►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 82 177 119)(2 63 178 100)(3 74 179 111)(4 85 180 92)(5 66 151 103)(6 77 152 114)(7 88 153 95)(8 69 154 106)(9 80 155 117)(10 61 156 98)(11 72 157 109)(12 83 158 120)(13 64 159 101)(14 75 160 112)(15 86 161 93)(16 67 162 104)(17 78 163 115)(18 89 164 96)(19 70 165 107)(20 81 166 118)(21 62 167 99)(22 73 168 110)(23 84 169 91)(24 65 170 102)(25 76 171 113)(26 87 172 94)(27 68 173 105)(28 79 174 116)(29 90 175 97)(30 71 176 108)(31 144 190 232)(32 125 191 213)(33 136 192 224)(34 147 193 235)(35 128 194 216)(36 139 195 227)(37 150 196 238)(38 131 197 219)(39 142 198 230)(40 123 199 211)(41 134 200 222)(42 145 201 233)(43 126 202 214)(44 137 203 225)(45 148 204 236)(46 129 205 217)(47 140 206 228)(48 121 207 239)(49 132 208 220)(50 143 209 231)(51 124 210 212)(52 135 181 223)(53 146 182 234)(54 127 183 215)(55 138 184 226)(56 149 185 237)(57 130 186 218)(58 141 187 229)(59 122 188 240)(60 133 189 221)
(1 135 162 238)(2 124 163 227)(3 143 164 216)(4 132 165 235)(5 121 166 224)(6 140 167 213)(7 129 168 232)(8 148 169 221)(9 137 170 240)(10 126 171 229)(11 145 172 218)(12 134 173 237)(13 123 174 226)(14 142 175 215)(15 131 176 234)(16 150 177 223)(17 139 178 212)(18 128 179 231)(19 147 180 220)(20 136 151 239)(21 125 152 228)(22 144 153 217)(23 133 154 236)(24 122 155 225)(25 141 156 214)(26 130 157 233)(27 149 158 222)(28 138 159 211)(29 127 160 230)(30 146 161 219)(31 110 205 88)(32 99 206 77)(33 118 207 66)(34 107 208 85)(35 96 209 74)(36 115 210 63)(37 104 181 82)(38 93 182 71)(39 112 183 90)(40 101 184 79)(41 120 185 68)(42 109 186 87)(43 98 187 76)(44 117 188 65)(45 106 189 84)(46 95 190 73)(47 114 191 62)(48 103 192 81)(49 92 193 70)(50 111 194 89)(51 100 195 78)(52 119 196 67)(53 108 197 86)(54 97 198 75)(55 116 199 64)(56 105 200 83)(57 94 201 72)(58 113 202 61)(59 102 203 80)(60 91 204 69)
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), (1,82,177,119)(2,63,178,100)(3,74,179,111)(4,85,180,92)(5,66,151,103)(6,77,152,114)(7,88,153,95)(8,69,154,106)(9,80,155,117)(10,61,156,98)(11,72,157,109)(12,83,158,120)(13,64,159,101)(14,75,160,112)(15,86,161,93)(16,67,162,104)(17,78,163,115)(18,89,164,96)(19,70,165,107)(20,81,166,118)(21,62,167,99)(22,73,168,110)(23,84,169,91)(24,65,170,102)(25,76,171,113)(26,87,172,94)(27,68,173,105)(28,79,174,116)(29,90,175,97)(30,71,176,108)(31,144,190,232)(32,125,191,213)(33,136,192,224)(34,147,193,235)(35,128,194,216)(36,139,195,227)(37,150,196,238)(38,131,197,219)(39,142,198,230)(40,123,199,211)(41,134,200,222)(42,145,201,233)(43,126,202,214)(44,137,203,225)(45,148,204,236)(46,129,205,217)(47,140,206,228)(48,121,207,239)(49,132,208,220)(50,143,209,231)(51,124,210,212)(52,135,181,223)(53,146,182,234)(54,127,183,215)(55,138,184,226)(56,149,185,237)(57,130,186,218)(58,141,187,229)(59,122,188,240)(60,133,189,221), (1,135,162,238)(2,124,163,227)(3,143,164,216)(4,132,165,235)(5,121,166,224)(6,140,167,213)(7,129,168,232)(8,148,169,221)(9,137,170,240)(10,126,171,229)(11,145,172,218)(12,134,173,237)(13,123,174,226)(14,142,175,215)(15,131,176,234)(16,150,177,223)(17,139,178,212)(18,128,179,231)(19,147,180,220)(20,136,151,239)(21,125,152,228)(22,144,153,217)(23,133,154,236)(24,122,155,225)(25,141,156,214)(26,130,157,233)(27,149,158,222)(28,138,159,211)(29,127,160,230)(30,146,161,219)(31,110,205,88)(32,99,206,77)(33,118,207,66)(34,107,208,85)(35,96,209,74)(36,115,210,63)(37,104,181,82)(38,93,182,71)(39,112,183,90)(40,101,184,79)(41,120,185,68)(42,109,186,87)(43,98,187,76)(44,117,188,65)(45,106,189,84)(46,95,190,73)(47,114,191,62)(48,103,192,81)(49,92,193,70)(50,111,194,89)(51,100,195,78)(52,119,196,67)(53,108,197,86)(54,97,198,75)(55,116,199,64)(56,105,200,83)(57,94,201,72)(58,113,202,61)(59,102,203,80)(60,91,204,69)>;
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), (1,82,177,119)(2,63,178,100)(3,74,179,111)(4,85,180,92)(5,66,151,103)(6,77,152,114)(7,88,153,95)(8,69,154,106)(9,80,155,117)(10,61,156,98)(11,72,157,109)(12,83,158,120)(13,64,159,101)(14,75,160,112)(15,86,161,93)(16,67,162,104)(17,78,163,115)(18,89,164,96)(19,70,165,107)(20,81,166,118)(21,62,167,99)(22,73,168,110)(23,84,169,91)(24,65,170,102)(25,76,171,113)(26,87,172,94)(27,68,173,105)(28,79,174,116)(29,90,175,97)(30,71,176,108)(31,144,190,232)(32,125,191,213)(33,136,192,224)(34,147,193,235)(35,128,194,216)(36,139,195,227)(37,150,196,238)(38,131,197,219)(39,142,198,230)(40,123,199,211)(41,134,200,222)(42,145,201,233)(43,126,202,214)(44,137,203,225)(45,148,204,236)(46,129,205,217)(47,140,206,228)(48,121,207,239)(49,132,208,220)(50,143,209,231)(51,124,210,212)(52,135,181,223)(53,146,182,234)(54,127,183,215)(55,138,184,226)(56,149,185,237)(57,130,186,218)(58,141,187,229)(59,122,188,240)(60,133,189,221), (1,135,162,238)(2,124,163,227)(3,143,164,216)(4,132,165,235)(5,121,166,224)(6,140,167,213)(7,129,168,232)(8,148,169,221)(9,137,170,240)(10,126,171,229)(11,145,172,218)(12,134,173,237)(13,123,174,226)(14,142,175,215)(15,131,176,234)(16,150,177,223)(17,139,178,212)(18,128,179,231)(19,147,180,220)(20,136,151,239)(21,125,152,228)(22,144,153,217)(23,133,154,236)(24,122,155,225)(25,141,156,214)(26,130,157,233)(27,149,158,222)(28,138,159,211)(29,127,160,230)(30,146,161,219)(31,110,205,88)(32,99,206,77)(33,118,207,66)(34,107,208,85)(35,96,209,74)(36,115,210,63)(37,104,181,82)(38,93,182,71)(39,112,183,90)(40,101,184,79)(41,120,185,68)(42,109,186,87)(43,98,187,76)(44,117,188,65)(45,106,189,84)(46,95,190,73)(47,114,191,62)(48,103,192,81)(49,92,193,70)(50,111,194,89)(51,100,195,78)(52,119,196,67)(53,108,197,86)(54,97,198,75)(55,116,199,64)(56,105,200,83)(57,94,201,72)(58,113,202,61)(59,102,203,80)(60,91,204,69) );
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)], [(1,82,177,119),(2,63,178,100),(3,74,179,111),(4,85,180,92),(5,66,151,103),(6,77,152,114),(7,88,153,95),(8,69,154,106),(9,80,155,117),(10,61,156,98),(11,72,157,109),(12,83,158,120),(13,64,159,101),(14,75,160,112),(15,86,161,93),(16,67,162,104),(17,78,163,115),(18,89,164,96),(19,70,165,107),(20,81,166,118),(21,62,167,99),(22,73,168,110),(23,84,169,91),(24,65,170,102),(25,76,171,113),(26,87,172,94),(27,68,173,105),(28,79,174,116),(29,90,175,97),(30,71,176,108),(31,144,190,232),(32,125,191,213),(33,136,192,224),(34,147,193,235),(35,128,194,216),(36,139,195,227),(37,150,196,238),(38,131,197,219),(39,142,198,230),(40,123,199,211),(41,134,200,222),(42,145,201,233),(43,126,202,214),(44,137,203,225),(45,148,204,236),(46,129,205,217),(47,140,206,228),(48,121,207,239),(49,132,208,220),(50,143,209,231),(51,124,210,212),(52,135,181,223),(53,146,182,234),(54,127,183,215),(55,138,184,226),(56,149,185,237),(57,130,186,218),(58,141,187,229),(59,122,188,240),(60,133,189,221)], [(1,135,162,238),(2,124,163,227),(3,143,164,216),(4,132,165,235),(5,121,166,224),(6,140,167,213),(7,129,168,232),(8,148,169,221),(9,137,170,240),(10,126,171,229),(11,145,172,218),(12,134,173,237),(13,123,174,226),(14,142,175,215),(15,131,176,234),(16,150,177,223),(17,139,178,212),(18,128,179,231),(19,147,180,220),(20,136,151,239),(21,125,152,228),(22,144,153,217),(23,133,154,236),(24,122,155,225),(25,141,156,214),(26,130,157,233),(27,149,158,222),(28,138,159,211),(29,127,160,230),(30,146,161,219),(31,110,205,88),(32,99,206,77),(33,118,207,66),(34,107,208,85),(35,96,209,74),(36,115,210,63),(37,104,181,82),(38,93,182,71),(39,112,183,90),(40,101,184,79),(41,120,185,68),(42,109,186,87),(43,98,187,76),(44,117,188,65),(45,106,189,84),(46,95,190,73),(47,114,191,62),(48,103,192,81),(49,92,193,70),(50,111,194,89),(51,100,195,78),(52,119,196,67),(53,108,197,86),(54,97,198,75),(55,116,199,64),(56,105,200,83),(57,94,201,72),(58,113,202,61),(59,102,203,80),(60,91,204,69)]])
C30.Q8 is a maximal subgroup of
Dic5⋊5Dic6 Dic3⋊Dic10 Dic15⋊Q8 Dic3×Dic10 Dic15⋊6Q8 Dic5.1Dic6 Dic5.2Dic6 Dic15.2Q8 D6⋊C4.D5 C4⋊Dic5⋊S3 Dic3.2Dic10 D10⋊Dic6 D30.34D4 (D5×C12)⋊C4 (C4×Dic3)⋊D5 C60.45D4 (C4×Dic15)⋊C2 D6⋊Dic5.C2 Dic5⋊Dic6 Dic5.7Dic6 Dic3.3Dic10 C10.D4⋊S3 C60.6Q8 Dic15.4Q8 D30⋊9Q8 C12.Dic10 S3×C10.D4 (S3×Dic5)⋊C4 D30.23(C2×C4) D30.Q8 Dic5⋊4D12 D6⋊1Dic10 D30⋊Q8 D5×C4⋊Dic3 D10.16D12 D10.17D12 D6⋊2Dic10 D30⋊2Q8 D10⋊1Dic6 D10⋊4Dic6 C4×C5⋊D12 D10⋊D12 D10⋊C4⋊S3 C4×C15⋊Q8 C60⋊Q8 (C6×Dic5)⋊7C4 C23.13(S3×D5) C23.14(S3×D5) D30⋊6D4 C10.(C2×D12) (C2×C10).D12 Dic3×C5⋊D4 (S3×C10).D4 Dic15⋊16D4 D30.16D4 (S3×C10)⋊D4 (C2×C10)⋊4D12 (C2×C30)⋊Q8 (C2×C10)⋊8Dic6 Dic15.48D4
C30.Q8 is a maximal quotient of
C60.13Q8 C60.7Q8 C60.8Q8 C60.105D4 C30.24C42
42 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 | 20A | ··· | 20H | 30A | ··· | 30F |
| 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 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | - | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D5 | Dic3 | D6 | D10 | Dic6 | D12 | Dic10 | C4×D5 | C5⋊D4 | S3×D5 | D5×Dic3 | C5⋊D12 | C15⋊Q8 |
| kernel | C30.Q8 | C6×Dic5 | C10×Dic3 | C2×Dic15 | C3×Dic5 | C2×Dic5 | C30 | C30 | C2×Dic3 | Dic5 | C2×C10 | C2×C6 | C10 | C10 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of C30.Q8 ►in GL4(𝔽61) generated by
| 0 | 60 | 0 | 0 |
| 1 | 18 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 60 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 17 | 25 |
| 0 | 0 | 8 | 44 |
| 33 | 6 | 0 | 0 |
| 22 | 28 | 0 | 0 |
| 0 | 0 | 38 | 15 |
| 0 | 0 | 46 | 23 |
G:=sub<GL(4,GF(61))| [0,1,0,0,60,18,0,0,0,0,1,60,0,0,1,0],[11,0,0,0,0,11,0,0,0,0,17,8,0,0,25,44],[33,22,0,0,6,28,0,0,0,0,38,46,0,0,15,23] >;
C30.Q8 in GAP, Magma, Sage, TeX
C_{30}.Q_8 % in TeX
G:=Group("C30.Q8"); // GroupNames label
G:=SmallGroup(240,29);
// by ID
G=gap.SmallGroup(240,29);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,121,31,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=1,c^2=a^15*b^2,b*a*b^-1=a^11,c*a*c^-1=a^19,c*b*c^-1=a^15*b^-1>;
// generators/relations
Export