Aliases: Q8.C54, C36.3A4, C4○D4⋊C27, Q8⋊C27⋊2C2, C9.(C4.A4), C4.(C9.A4), (Q8×C9).6C6, C18.12(C2×A4), (C3×Q8).3C18, C3.(Q8.C18), C12.2(C3.A4), (C3×C4○D4).C9, (C9×C4○D4).C3, C2.3(C2×C9.A4), C6.6(C2×C3.A4), SmallGroup(432,42)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — Q8.C54 |
Generators and relations for Q8.C54
G = < a,b,c | a4=1, b2=c54=a2, bab-1=a-1, cac-1=ab, cbc-1=a >
Smallest permutation representation of Q8.C54
►On 216 points
Generators in S216
(1 175 55 121)(2 149 56 203)(3 30 57 84)(4 178 58 124)(5 152 59 206)(6 33 60 87)(7 181 61 127)(8 155 62 209)(9 36 63 90)(10 184 64 130)(11 158 65 212)(12 39 66 93)(13 187 67 133)(14 161 68 215)(15 42 69 96)(16 190 70 136)(17 164 71 110)(18 45 72 99)(19 193 73 139)(20 167 74 113)(21 48 75 102)(22 196 76 142)(23 170 77 116)(24 51 78 105)(25 199 79 145)(26 173 80 119)(27 54 81 108)(28 202 82 148)(29 176 83 122)(31 205 85 151)(32 179 86 125)(34 208 88 154)(35 182 89 128)(37 211 91 157)(38 185 92 131)(40 214 94 160)(41 188 95 134)(43 109 97 163)(44 191 98 137)(46 112 100 166)(47 194 101 140)(49 115 103 169)(50 197 104 143)(52 118 106 172)(53 200 107 146)(111 192 165 138)(114 195 168 141)(117 198 171 144)(120 201 174 147)(123 204 177 150)(126 207 180 153)(129 210 183 156)(132 213 186 159)(135 216 189 162)
(1 28 55 82)(2 176 56 122)(3 150 57 204)(4 31 58 85)(5 179 59 125)(6 153 60 207)(7 34 61 88)(8 182 62 128)(9 156 63 210)(10 37 64 91)(11 185 65 131)(12 159 66 213)(13 40 67 94)(14 188 68 134)(15 162 69 216)(16 43 70 97)(17 191 71 137)(18 165 72 111)(19 46 73 100)(20 194 74 140)(21 168 75 114)(22 49 76 103)(23 197 77 143)(24 171 78 117)(25 52 79 106)(26 200 80 146)(27 174 81 120)(29 203 83 149)(30 177 84 123)(32 206 86 152)(33 180 87 126)(35 209 89 155)(36 183 90 129)(38 212 92 158)(39 186 93 132)(41 215 95 161)(42 189 96 135)(44 110 98 164)(45 192 99 138)(47 113 101 167)(48 195 102 141)(50 116 104 170)(51 198 105 144)(53 119 107 173)(54 201 108 147)(109 190 163 136)(112 193 166 139)(115 196 169 142)(118 199 172 145)(121 202 175 148)(124 205 178 151)(127 208 181 154)(130 211 184 157)(133 214 187 160)
(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)
G:=sub<Sym(216)| (1,175,55,121)(2,149,56,203)(3,30,57,84)(4,178,58,124)(5,152,59,206)(6,33,60,87)(7,181,61,127)(8,155,62,209)(9,36,63,90)(10,184,64,130)(11,158,65,212)(12,39,66,93)(13,187,67,133)(14,161,68,215)(15,42,69,96)(16,190,70,136)(17,164,71,110)(18,45,72,99)(19,193,73,139)(20,167,74,113)(21,48,75,102)(22,196,76,142)(23,170,77,116)(24,51,78,105)(25,199,79,145)(26,173,80,119)(27,54,81,108)(28,202,82,148)(29,176,83,122)(31,205,85,151)(32,179,86,125)(34,208,88,154)(35,182,89,128)(37,211,91,157)(38,185,92,131)(40,214,94,160)(41,188,95,134)(43,109,97,163)(44,191,98,137)(46,112,100,166)(47,194,101,140)(49,115,103,169)(50,197,104,143)(52,118,106,172)(53,200,107,146)(111,192,165,138)(114,195,168,141)(117,198,171,144)(120,201,174,147)(123,204,177,150)(126,207,180,153)(129,210,183,156)(132,213,186,159)(135,216,189,162), (1,28,55,82)(2,176,56,122)(3,150,57,204)(4,31,58,85)(5,179,59,125)(6,153,60,207)(7,34,61,88)(8,182,62,128)(9,156,63,210)(10,37,64,91)(11,185,65,131)(12,159,66,213)(13,40,67,94)(14,188,68,134)(15,162,69,216)(16,43,70,97)(17,191,71,137)(18,165,72,111)(19,46,73,100)(20,194,74,140)(21,168,75,114)(22,49,76,103)(23,197,77,143)(24,171,78,117)(25,52,79,106)(26,200,80,146)(27,174,81,120)(29,203,83,149)(30,177,84,123)(32,206,86,152)(33,180,87,126)(35,209,89,155)(36,183,90,129)(38,212,92,158)(39,186,93,132)(41,215,95,161)(42,189,96,135)(44,110,98,164)(45,192,99,138)(47,113,101,167)(48,195,102,141)(50,116,104,170)(51,198,105,144)(53,119,107,173)(54,201,108,147)(109,190,163,136)(112,193,166,139)(115,196,169,142)(118,199,172,145)(121,202,175,148)(124,205,178,151)(127,208,181,154)(130,211,184,157)(133,214,187,160), (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)>;
G:=Group( (1,175,55,121)(2,149,56,203)(3,30,57,84)(4,178,58,124)(5,152,59,206)(6,33,60,87)(7,181,61,127)(8,155,62,209)(9,36,63,90)(10,184,64,130)(11,158,65,212)(12,39,66,93)(13,187,67,133)(14,161,68,215)(15,42,69,96)(16,190,70,136)(17,164,71,110)(18,45,72,99)(19,193,73,139)(20,167,74,113)(21,48,75,102)(22,196,76,142)(23,170,77,116)(24,51,78,105)(25,199,79,145)(26,173,80,119)(27,54,81,108)(28,202,82,148)(29,176,83,122)(31,205,85,151)(32,179,86,125)(34,208,88,154)(35,182,89,128)(37,211,91,157)(38,185,92,131)(40,214,94,160)(41,188,95,134)(43,109,97,163)(44,191,98,137)(46,112,100,166)(47,194,101,140)(49,115,103,169)(50,197,104,143)(52,118,106,172)(53,200,107,146)(111,192,165,138)(114,195,168,141)(117,198,171,144)(120,201,174,147)(123,204,177,150)(126,207,180,153)(129,210,183,156)(132,213,186,159)(135,216,189,162), (1,28,55,82)(2,176,56,122)(3,150,57,204)(4,31,58,85)(5,179,59,125)(6,153,60,207)(7,34,61,88)(8,182,62,128)(9,156,63,210)(10,37,64,91)(11,185,65,131)(12,159,66,213)(13,40,67,94)(14,188,68,134)(15,162,69,216)(16,43,70,97)(17,191,71,137)(18,165,72,111)(19,46,73,100)(20,194,74,140)(21,168,75,114)(22,49,76,103)(23,197,77,143)(24,171,78,117)(25,52,79,106)(26,200,80,146)(27,174,81,120)(29,203,83,149)(30,177,84,123)(32,206,86,152)(33,180,87,126)(35,209,89,155)(36,183,90,129)(38,212,92,158)(39,186,93,132)(41,215,95,161)(42,189,96,135)(44,110,98,164)(45,192,99,138)(47,113,101,167)(48,195,102,141)(50,116,104,170)(51,198,105,144)(53,119,107,173)(54,201,108,147)(109,190,163,136)(112,193,166,139)(115,196,169,142)(118,199,172,145)(121,202,175,148)(124,205,178,151)(127,208,181,154)(130,211,184,157)(133,214,187,160), (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) );
G=PermutationGroup([[(1,175,55,121),(2,149,56,203),(3,30,57,84),(4,178,58,124),(5,152,59,206),(6,33,60,87),(7,181,61,127),(8,155,62,209),(9,36,63,90),(10,184,64,130),(11,158,65,212),(12,39,66,93),(13,187,67,133),(14,161,68,215),(15,42,69,96),(16,190,70,136),(17,164,71,110),(18,45,72,99),(19,193,73,139),(20,167,74,113),(21,48,75,102),(22,196,76,142),(23,170,77,116),(24,51,78,105),(25,199,79,145),(26,173,80,119),(27,54,81,108),(28,202,82,148),(29,176,83,122),(31,205,85,151),(32,179,86,125),(34,208,88,154),(35,182,89,128),(37,211,91,157),(38,185,92,131),(40,214,94,160),(41,188,95,134),(43,109,97,163),(44,191,98,137),(46,112,100,166),(47,194,101,140),(49,115,103,169),(50,197,104,143),(52,118,106,172),(53,200,107,146),(111,192,165,138),(114,195,168,141),(117,198,171,144),(120,201,174,147),(123,204,177,150),(126,207,180,153),(129,210,183,156),(132,213,186,159),(135,216,189,162)], [(1,28,55,82),(2,176,56,122),(3,150,57,204),(4,31,58,85),(5,179,59,125),(6,153,60,207),(7,34,61,88),(8,182,62,128),(9,156,63,210),(10,37,64,91),(11,185,65,131),(12,159,66,213),(13,40,67,94),(14,188,68,134),(15,162,69,216),(16,43,70,97),(17,191,71,137),(18,165,72,111),(19,46,73,100),(20,194,74,140),(21,168,75,114),(22,49,76,103),(23,197,77,143),(24,171,78,117),(25,52,79,106),(26,200,80,146),(27,174,81,120),(29,203,83,149),(30,177,84,123),(32,206,86,152),(33,180,87,126),(35,209,89,155),(36,183,90,129),(38,212,92,158),(39,186,93,132),(41,215,95,161),(42,189,96,135),(44,110,98,164),(45,192,99,138),(47,113,101,167),(48,195,102,141),(50,116,104,170),(51,198,105,144),(53,119,107,173),(54,201,108,147),(109,190,163,136),(112,193,166,139),(115,196,169,142),(118,199,172,145),(121,202,175,148),(124,205,178,151),(127,208,181,154),(130,211,184,157),(133,214,187,160)], [(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)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | ··· | 18F | 18G | ··· | 18L | 27A | ··· | 27R | 36A | ··· | 36L | 36M | ··· | 36R | 54A | ··· | 54R | 108A | ··· | 108AJ |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 6 | 1 | 1 | 1 | 1 | 6 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 6 | ··· | 6 | 4 | ··· | 4 | 1 | ··· | 1 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | |||||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | C4.A4 | Q8.C18 | Q8.C54 | A4 | C2×A4 | C3.A4 | C2×C3.A4 | C9.A4 | C2×C9.A4 |
| kernel | Q8.C54 | Q8⋊C27 | C9×C4○D4 | Q8×C9 | C3×C4○D4 | C3×Q8 | C4○D4 | Q8 | C9 | C3 | C1 | C36 | C18 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 6 | 12 | 36 | 1 | 1 | 2 | 2 | 6 | 6 |
Matrix representation of Q8.C54 ►in GL2(𝔽109) generated by
| 0 | 108 |
| 1 | 0 |
| 76 | 0 |
| 0 | 33 |
| 28 | 57 |
| 81 | 57 |
G:=sub<GL(2,GF(109))| [0,1,108,0],[76,0,0,33],[28,81,57,57] >;
Q8.C54 in GAP, Magma, Sage, TeX
Q_8.C_{54} % in TeX
G:=Group("Q8.C54"); // GroupNames label
G:=SmallGroup(432,42);
// by ID
G=gap.SmallGroup(432,42);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,50,79,1901,172,3414,285,124]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^2=c^54=a^2,b*a*b^-1=a^-1,c*a*c^-1=a*b,c*b*c^-1=a>;
// generators/relations
Export