Aliases: Q8⋊C27, C18.2A4, C9.SL2(𝔽3), C3.(Q8⋊C9), (Q8×C9).C3, (C3×Q8).C9, C2.(C9.A4), C6.1(C3.A4), SmallGroup(216,3)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — Q8⋊C27 |
Generators and relations for Q8⋊C27
G = < a,b,c | a4=c27=1, b2=a2, bab-1=a-1, cac-1=b, cbc-1=ab >
Smallest permutation representation of Q8⋊C27
►Regular action on 216 points
Generators in S216
(1 184 132 139)(2 191 133 56)(3 107 134 40)(4 187 135 142)(5 194 109 59)(6 83 110 43)(7 163 111 145)(8 197 112 62)(9 86 113 46)(10 166 114 148)(11 200 115 65)(12 89 116 49)(13 169 117 151)(14 203 118 68)(15 92 119 52)(16 172 120 154)(17 206 121 71)(18 95 122 28)(19 175 123 157)(20 209 124 74)(21 98 125 31)(22 178 126 160)(23 212 127 77)(24 101 128 34)(25 181 129 136)(26 215 130 80)(27 104 131 37)(29 73 96 208)(30 176 97 158)(32 76 99 211)(33 179 100 161)(35 79 102 214)(36 182 103 137)(38 55 105 190)(39 185 106 140)(41 58 108 193)(42 188 82 143)(44 61 84 196)(45 164 85 146)(47 64 87 199)(48 167 88 149)(50 67 90 202)(51 170 91 152)(53 70 93 205)(54 173 94 155)(57 141 192 186)(60 144 195 189)(63 147 198 165)(66 150 201 168)(69 153 204 171)(72 156 207 174)(75 159 210 177)(78 162 213 180)(81 138 216 183)
(1 190 132 55)(2 106 133 39)(3 186 134 141)(4 193 135 58)(5 82 109 42)(6 189 110 144)(7 196 111 61)(8 85 112 45)(9 165 113 147)(10 199 114 64)(11 88 115 48)(12 168 116 150)(13 202 117 67)(14 91 118 51)(15 171 119 153)(16 205 120 70)(17 94 121 54)(18 174 122 156)(19 208 123 73)(20 97 124 30)(21 177 125 159)(22 211 126 76)(23 100 127 33)(24 180 128 162)(25 214 129 79)(26 103 130 36)(27 183 131 138)(28 72 95 207)(29 175 96 157)(31 75 98 210)(32 178 99 160)(34 78 101 213)(35 181 102 136)(37 81 104 216)(38 184 105 139)(40 57 107 192)(41 187 108 142)(43 60 83 195)(44 163 84 145)(46 63 86 198)(47 166 87 148)(49 66 89 201)(50 169 90 151)(52 69 92 204)(53 172 93 154)(56 140 191 185)(59 143 194 188)(62 146 197 164)(65 149 200 167)(68 152 203 170)(71 155 206 173)(74 158 209 176)(77 161 212 179)(80 137 215 182)
(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)| (1,184,132,139)(2,191,133,56)(3,107,134,40)(4,187,135,142)(5,194,109,59)(6,83,110,43)(7,163,111,145)(8,197,112,62)(9,86,113,46)(10,166,114,148)(11,200,115,65)(12,89,116,49)(13,169,117,151)(14,203,118,68)(15,92,119,52)(16,172,120,154)(17,206,121,71)(18,95,122,28)(19,175,123,157)(20,209,124,74)(21,98,125,31)(22,178,126,160)(23,212,127,77)(24,101,128,34)(25,181,129,136)(26,215,130,80)(27,104,131,37)(29,73,96,208)(30,176,97,158)(32,76,99,211)(33,179,100,161)(35,79,102,214)(36,182,103,137)(38,55,105,190)(39,185,106,140)(41,58,108,193)(42,188,82,143)(44,61,84,196)(45,164,85,146)(47,64,87,199)(48,167,88,149)(50,67,90,202)(51,170,91,152)(53,70,93,205)(54,173,94,155)(57,141,192,186)(60,144,195,189)(63,147,198,165)(66,150,201,168)(69,153,204,171)(72,156,207,174)(75,159,210,177)(78,162,213,180)(81,138,216,183), (1,190,132,55)(2,106,133,39)(3,186,134,141)(4,193,135,58)(5,82,109,42)(6,189,110,144)(7,196,111,61)(8,85,112,45)(9,165,113,147)(10,199,114,64)(11,88,115,48)(12,168,116,150)(13,202,117,67)(14,91,118,51)(15,171,119,153)(16,205,120,70)(17,94,121,54)(18,174,122,156)(19,208,123,73)(20,97,124,30)(21,177,125,159)(22,211,126,76)(23,100,127,33)(24,180,128,162)(25,214,129,79)(26,103,130,36)(27,183,131,138)(28,72,95,207)(29,175,96,157)(31,75,98,210)(32,178,99,160)(34,78,101,213)(35,181,102,136)(37,81,104,216)(38,184,105,139)(40,57,107,192)(41,187,108,142)(43,60,83,195)(44,163,84,145)(46,63,86,198)(47,166,87,148)(49,66,89,201)(50,169,90,151)(52,69,92,204)(53,172,93,154)(56,140,191,185)(59,143,194,188)(62,146,197,164)(65,149,200,167)(68,152,203,170)(71,155,206,173)(74,158,209,176)(77,161,212,179)(80,137,215,182), (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( (1,184,132,139)(2,191,133,56)(3,107,134,40)(4,187,135,142)(5,194,109,59)(6,83,110,43)(7,163,111,145)(8,197,112,62)(9,86,113,46)(10,166,114,148)(11,200,115,65)(12,89,116,49)(13,169,117,151)(14,203,118,68)(15,92,119,52)(16,172,120,154)(17,206,121,71)(18,95,122,28)(19,175,123,157)(20,209,124,74)(21,98,125,31)(22,178,126,160)(23,212,127,77)(24,101,128,34)(25,181,129,136)(26,215,130,80)(27,104,131,37)(29,73,96,208)(30,176,97,158)(32,76,99,211)(33,179,100,161)(35,79,102,214)(36,182,103,137)(38,55,105,190)(39,185,106,140)(41,58,108,193)(42,188,82,143)(44,61,84,196)(45,164,85,146)(47,64,87,199)(48,167,88,149)(50,67,90,202)(51,170,91,152)(53,70,93,205)(54,173,94,155)(57,141,192,186)(60,144,195,189)(63,147,198,165)(66,150,201,168)(69,153,204,171)(72,156,207,174)(75,159,210,177)(78,162,213,180)(81,138,216,183), (1,190,132,55)(2,106,133,39)(3,186,134,141)(4,193,135,58)(5,82,109,42)(6,189,110,144)(7,196,111,61)(8,85,112,45)(9,165,113,147)(10,199,114,64)(11,88,115,48)(12,168,116,150)(13,202,117,67)(14,91,118,51)(15,171,119,153)(16,205,120,70)(17,94,121,54)(18,174,122,156)(19,208,123,73)(20,97,124,30)(21,177,125,159)(22,211,126,76)(23,100,127,33)(24,180,128,162)(25,214,129,79)(26,103,130,36)(27,183,131,138)(28,72,95,207)(29,175,96,157)(31,75,98,210)(32,178,99,160)(34,78,101,213)(35,181,102,136)(37,81,104,216)(38,184,105,139)(40,57,107,192)(41,187,108,142)(43,60,83,195)(44,163,84,145)(46,63,86,198)(47,166,87,148)(49,66,89,201)(50,169,90,151)(52,69,92,204)(53,172,93,154)(56,140,191,185)(59,143,194,188)(62,146,197,164)(65,149,200,167)(68,152,203,170)(71,155,206,173)(74,158,209,176)(77,161,212,179)(80,137,215,182), (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([[(1,184,132,139),(2,191,133,56),(3,107,134,40),(4,187,135,142),(5,194,109,59),(6,83,110,43),(7,163,111,145),(8,197,112,62),(9,86,113,46),(10,166,114,148),(11,200,115,65),(12,89,116,49),(13,169,117,151),(14,203,118,68),(15,92,119,52),(16,172,120,154),(17,206,121,71),(18,95,122,28),(19,175,123,157),(20,209,124,74),(21,98,125,31),(22,178,126,160),(23,212,127,77),(24,101,128,34),(25,181,129,136),(26,215,130,80),(27,104,131,37),(29,73,96,208),(30,176,97,158),(32,76,99,211),(33,179,100,161),(35,79,102,214),(36,182,103,137),(38,55,105,190),(39,185,106,140),(41,58,108,193),(42,188,82,143),(44,61,84,196),(45,164,85,146),(47,64,87,199),(48,167,88,149),(50,67,90,202),(51,170,91,152),(53,70,93,205),(54,173,94,155),(57,141,192,186),(60,144,195,189),(63,147,198,165),(66,150,201,168),(69,153,204,171),(72,156,207,174),(75,159,210,177),(78,162,213,180),(81,138,216,183)], [(1,190,132,55),(2,106,133,39),(3,186,134,141),(4,193,135,58),(5,82,109,42),(6,189,110,144),(7,196,111,61),(8,85,112,45),(9,165,113,147),(10,199,114,64),(11,88,115,48),(12,168,116,150),(13,202,117,67),(14,91,118,51),(15,171,119,153),(16,205,120,70),(17,94,121,54),(18,174,122,156),(19,208,123,73),(20,97,124,30),(21,177,125,159),(22,211,126,76),(23,100,127,33),(24,180,128,162),(25,214,129,79),(26,103,130,36),(27,183,131,138),(28,72,95,207),(29,175,96,157),(31,75,98,210),(32,178,99,160),(34,78,101,213),(35,181,102,136),(37,81,104,216),(38,184,105,139),(40,57,107,192),(41,187,108,142),(43,60,83,195),(44,163,84,145),(46,63,86,198),(47,166,87,148),(49,66,89,201),(50,169,90,151),(52,69,92,204),(53,172,93,154),(56,140,191,185),(59,143,194,188),(62,146,197,164),(65,149,200,167),(68,152,203,170),(71,155,206,173),(74,158,209,176),(77,161,212,179),(80,137,215,182)], [(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)]])
Q8⋊C27 is a maximal subgroup of
Q8.D27 Q8⋊D27 Q8.C54
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4 | 6A | 6B | 9A | ··· | 9F | 12A | 12B | 18A | ··· | 18F | 27A | ··· | 27R | 36A | ··· | 36F | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 4 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | 1 | 6 | 1 | 1 | 1 | ··· | 1 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 |
| type | + | - | + | ||||||||
| image | C1 | C3 | C9 | C27 | SL2(𝔽3) | SL2(𝔽3) | Q8⋊C9 | Q8⋊C27 | A4 | C3.A4 | C9.A4 |
| kernel | Q8⋊C27 | Q8×C9 | C3×Q8 | Q8 | C9 | C9 | C3 | C1 | C18 | C6 | C2 |
| # reps | 1 | 2 | 6 | 18 | 1 | 2 | 6 | 18 | 1 | 2 | 6 |
Matrix representation of Q8⋊C27 ►in GL2(𝔽109) generated by
| 1 | 2 |
| 108 | 108 |
| 38 | 63 |
| 48 | 71 |
| 76 | 90 |
| 78 | 94 |
G:=sub<GL(2,GF(109))| [1,108,2,108],[38,48,63,71],[76,78,90,94] >;
Q8⋊C27 in GAP, Magma, Sage, TeX
Q_8\rtimes C_{27} % in TeX
G:=Group("Q8:C27"); // GroupNames label
G:=SmallGroup(216,3);
// by ID
G=gap.SmallGroup(216,3);
# by ID
G:=PCGroup([6,-3,-3,-3,-2,2,-2,18,43,1299,117,2434,202,88]);
// Polycyclic
G:=Group<a,b,c|a^4=c^27=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,c*b*c^-1=a*b>;
// generators/relations
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