direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C27, C4.C54, C36.8C6, C12.4C18, C108.3C2, C54.7C22, C9.(C3×Q8), C3.(Q8×C9), C54○(Q8×C9), C2.2(C2×C54), C6.7(C2×C18), (Q8×C9).2C3, (C3×Q8).2C9, C18.15(C2×C6), SmallGroup(216,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C27
G = < a,b,c | a27=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C27
►Regular action on 216 points
Generators in S216
(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)
(1 121 170 49)(2 122 171 50)(3 123 172 51)(4 124 173 52)(5 125 174 53)(6 126 175 54)(7 127 176 28)(8 128 177 29)(9 129 178 30)(10 130 179 31)(11 131 180 32)(12 132 181 33)(13 133 182 34)(14 134 183 35)(15 135 184 36)(16 109 185 37)(17 110 186 38)(18 111 187 39)(19 112 188 40)(20 113 189 41)(21 114 163 42)(22 115 164 43)(23 116 165 44)(24 117 166 45)(25 118 167 46)(26 119 168 47)(27 120 169 48)(55 201 159 96)(56 202 160 97)(57 203 161 98)(58 204 162 99)(59 205 136 100)(60 206 137 101)(61 207 138 102)(62 208 139 103)(63 209 140 104)(64 210 141 105)(65 211 142 106)(66 212 143 107)(67 213 144 108)(68 214 145 82)(69 215 146 83)(70 216 147 84)(71 190 148 85)(72 191 149 86)(73 192 150 87)(74 193 151 88)(75 194 152 89)(76 195 153 90)(77 196 154 91)(78 197 155 92)(79 198 156 93)(80 199 157 94)(81 200 158 95)
(1 62 170 139)(2 63 171 140)(3 64 172 141)(4 65 173 142)(5 66 174 143)(6 67 175 144)(7 68 176 145)(8 69 177 146)(9 70 178 147)(10 71 179 148)(11 72 180 149)(12 73 181 150)(13 74 182 151)(14 75 183 152)(15 76 184 153)(16 77 185 154)(17 78 186 155)(18 79 187 156)(19 80 188 157)(20 81 189 158)(21 55 163 159)(22 56 164 160)(23 57 165 161)(24 58 166 162)(25 59 167 136)(26 60 168 137)(27 61 169 138)(28 214 127 82)(29 215 128 83)(30 216 129 84)(31 190 130 85)(32 191 131 86)(33 192 132 87)(34 193 133 88)(35 194 134 89)(36 195 135 90)(37 196 109 91)(38 197 110 92)(39 198 111 93)(40 199 112 94)(41 200 113 95)(42 201 114 96)(43 202 115 97)(44 203 116 98)(45 204 117 99)(46 205 118 100)(47 206 119 101)(48 207 120 102)(49 208 121 103)(50 209 122 104)(51 210 123 105)(52 211 124 106)(53 212 125 107)(54 213 126 108)
G:=sub<Sym(216)| (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), (1,121,170,49)(2,122,171,50)(3,123,172,51)(4,124,173,52)(5,125,174,53)(6,126,175,54)(7,127,176,28)(8,128,177,29)(9,129,178,30)(10,130,179,31)(11,131,180,32)(12,132,181,33)(13,133,182,34)(14,134,183,35)(15,135,184,36)(16,109,185,37)(17,110,186,38)(18,111,187,39)(19,112,188,40)(20,113,189,41)(21,114,163,42)(22,115,164,43)(23,116,165,44)(24,117,166,45)(25,118,167,46)(26,119,168,47)(27,120,169,48)(55,201,159,96)(56,202,160,97)(57,203,161,98)(58,204,162,99)(59,205,136,100)(60,206,137,101)(61,207,138,102)(62,208,139,103)(63,209,140,104)(64,210,141,105)(65,211,142,106)(66,212,143,107)(67,213,144,108)(68,214,145,82)(69,215,146,83)(70,216,147,84)(71,190,148,85)(72,191,149,86)(73,192,150,87)(74,193,151,88)(75,194,152,89)(76,195,153,90)(77,196,154,91)(78,197,155,92)(79,198,156,93)(80,199,157,94)(81,200,158,95), (1,62,170,139)(2,63,171,140)(3,64,172,141)(4,65,173,142)(5,66,174,143)(6,67,175,144)(7,68,176,145)(8,69,177,146)(9,70,178,147)(10,71,179,148)(11,72,180,149)(12,73,181,150)(13,74,182,151)(14,75,183,152)(15,76,184,153)(16,77,185,154)(17,78,186,155)(18,79,187,156)(19,80,188,157)(20,81,189,158)(21,55,163,159)(22,56,164,160)(23,57,165,161)(24,58,166,162)(25,59,167,136)(26,60,168,137)(27,61,169,138)(28,214,127,82)(29,215,128,83)(30,216,129,84)(31,190,130,85)(32,191,131,86)(33,192,132,87)(34,193,133,88)(35,194,134,89)(36,195,135,90)(37,196,109,91)(38,197,110,92)(39,198,111,93)(40,199,112,94)(41,200,113,95)(42,201,114,96)(43,202,115,97)(44,203,116,98)(45,204,117,99)(46,205,118,100)(47,206,119,101)(48,207,120,102)(49,208,121,103)(50,209,122,104)(51,210,123,105)(52,211,124,106)(53,212,125,107)(54,213,126,108)>;
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), (1,121,170,49)(2,122,171,50)(3,123,172,51)(4,124,173,52)(5,125,174,53)(6,126,175,54)(7,127,176,28)(8,128,177,29)(9,129,178,30)(10,130,179,31)(11,131,180,32)(12,132,181,33)(13,133,182,34)(14,134,183,35)(15,135,184,36)(16,109,185,37)(17,110,186,38)(18,111,187,39)(19,112,188,40)(20,113,189,41)(21,114,163,42)(22,115,164,43)(23,116,165,44)(24,117,166,45)(25,118,167,46)(26,119,168,47)(27,120,169,48)(55,201,159,96)(56,202,160,97)(57,203,161,98)(58,204,162,99)(59,205,136,100)(60,206,137,101)(61,207,138,102)(62,208,139,103)(63,209,140,104)(64,210,141,105)(65,211,142,106)(66,212,143,107)(67,213,144,108)(68,214,145,82)(69,215,146,83)(70,216,147,84)(71,190,148,85)(72,191,149,86)(73,192,150,87)(74,193,151,88)(75,194,152,89)(76,195,153,90)(77,196,154,91)(78,197,155,92)(79,198,156,93)(80,199,157,94)(81,200,158,95), (1,62,170,139)(2,63,171,140)(3,64,172,141)(4,65,173,142)(5,66,174,143)(6,67,175,144)(7,68,176,145)(8,69,177,146)(9,70,178,147)(10,71,179,148)(11,72,180,149)(12,73,181,150)(13,74,182,151)(14,75,183,152)(15,76,184,153)(16,77,185,154)(17,78,186,155)(18,79,187,156)(19,80,188,157)(20,81,189,158)(21,55,163,159)(22,56,164,160)(23,57,165,161)(24,58,166,162)(25,59,167,136)(26,60,168,137)(27,61,169,138)(28,214,127,82)(29,215,128,83)(30,216,129,84)(31,190,130,85)(32,191,131,86)(33,192,132,87)(34,193,133,88)(35,194,134,89)(36,195,135,90)(37,196,109,91)(38,197,110,92)(39,198,111,93)(40,199,112,94)(41,200,113,95)(42,201,114,96)(43,202,115,97)(44,203,116,98)(45,204,117,99)(46,205,118,100)(47,206,119,101)(48,207,120,102)(49,208,121,103)(50,209,122,104)(51,210,123,105)(52,211,124,106)(53,212,125,107)(54,213,126,108) );
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)], [(1,121,170,49),(2,122,171,50),(3,123,172,51),(4,124,173,52),(5,125,174,53),(6,126,175,54),(7,127,176,28),(8,128,177,29),(9,129,178,30),(10,130,179,31),(11,131,180,32),(12,132,181,33),(13,133,182,34),(14,134,183,35),(15,135,184,36),(16,109,185,37),(17,110,186,38),(18,111,187,39),(19,112,188,40),(20,113,189,41),(21,114,163,42),(22,115,164,43),(23,116,165,44),(24,117,166,45),(25,118,167,46),(26,119,168,47),(27,120,169,48),(55,201,159,96),(56,202,160,97),(57,203,161,98),(58,204,162,99),(59,205,136,100),(60,206,137,101),(61,207,138,102),(62,208,139,103),(63,209,140,104),(64,210,141,105),(65,211,142,106),(66,212,143,107),(67,213,144,108),(68,214,145,82),(69,215,146,83),(70,216,147,84),(71,190,148,85),(72,191,149,86),(73,192,150,87),(74,193,151,88),(75,194,152,89),(76,195,153,90),(77,196,154,91),(78,197,155,92),(79,198,156,93),(80,199,157,94),(81,200,158,95)], [(1,62,170,139),(2,63,171,140),(3,64,172,141),(4,65,173,142),(5,66,174,143),(6,67,175,144),(7,68,176,145),(8,69,177,146),(9,70,178,147),(10,71,179,148),(11,72,180,149),(12,73,181,150),(13,74,182,151),(14,75,183,152),(15,76,184,153),(16,77,185,154),(17,78,186,155),(18,79,187,156),(19,80,188,157),(20,81,189,158),(21,55,163,159),(22,56,164,160),(23,57,165,161),(24,58,166,162),(25,59,167,136),(26,60,168,137),(27,61,169,138),(28,214,127,82),(29,215,128,83),(30,216,129,84),(31,190,130,85),(32,191,131,86),(33,192,132,87),(34,193,133,88),(35,194,134,89),(36,195,135,90),(37,196,109,91),(38,197,110,92),(39,198,111,93),(40,199,112,94),(41,200,113,95),(42,201,114,96),(43,202,115,97),(44,203,116,98),(45,204,117,99),(46,205,118,100),(47,206,119,101),(48,207,120,102),(49,208,121,103),(50,209,122,104),(51,210,123,105),(52,211,124,106),(53,212,125,107),(54,213,126,108)]])
Q8×C27 is a maximal subgroup of
C27⋊Q16 Q8⋊2D27 Q8⋊3D27
135 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 9A | ··· | 9F | 12A | ··· | 12F | 18A | ··· | 18F | 27A | ··· | 27R | 36A | ··· | 36R | 54A | ··· | 54R | 108A | ··· | 108BB |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | Q8 | C3×Q8 | Q8×C9 | Q8×C27 |
| kernel | Q8×C27 | C108 | Q8×C9 | C36 | C3×Q8 | C12 | Q8 | C4 | C27 | C9 | C3 | C1 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 | 18 | 54 | 1 | 2 | 6 | 18 |
Matrix representation of Q8×C27 ►in GL3(𝔽109) generated by
| 97 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 108 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 108 | 0 |
| 1 | 0 | 0 |
| 0 | 67 | 57 |
| 0 | 57 | 42 |
G:=sub<GL(3,GF(109))| [97,0,0,0,16,0,0,0,16],[108,0,0,0,0,108,0,1,0],[1,0,0,0,67,57,0,57,42] >;
Q8×C27 in GAP, Magma, Sage, TeX
Q_8\times C_{27} % in TeX
G:=Group("Q8xC27"); // GroupNames label
G:=SmallGroup(216,11);
// by ID
G=gap.SmallGroup(216,11);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,122,118]);
// Polycyclic
G:=Group<a,b,c|a^27=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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