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G = Q8⋊C27order 216 = 23·33

The semidirect product of Q8 and C27 acting via C27/C9=C3

non-abelian, soluble

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

C1C2Q8 — Q8⋊C27
C1C2Q8C3×Q8Q8×C9 — Q8⋊C27
Q8 — Q8⋊C27
C1C18

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 >

3C4
3C12
4C27
3C36
4C54

Smallest permutation representation of Q8⋊C27
Regular action on 216 points
Generators in S216
(1 127 97 150)(2 34 98 170)(3 196 99 61)(4 130 100 153)(5 37 101 173)(6 199 102 64)(7 133 103 156)(8 40 104 176)(9 202 105 67)(10 109 106 159)(11 43 107 179)(12 205 108 70)(13 112 82 162)(14 46 83 182)(15 208 84 73)(16 115 85 138)(17 49 86 185)(18 211 87 76)(19 118 88 141)(20 52 89 188)(21 214 90 79)(22 121 91 144)(23 28 92 164)(24 190 93 55)(25 124 94 147)(26 31 95 167)(27 193 96 58)(29 123 165 146)(30 56 166 191)(32 126 168 149)(33 59 169 194)(35 129 171 152)(36 62 172 197)(38 132 174 155)(39 65 175 200)(41 135 177 158)(42 68 178 203)(44 111 180 161)(45 71 181 206)(47 114 183 137)(48 74 184 209)(50 117 186 140)(51 77 187 212)(53 120 189 143)(54 80 163 215)(57 125 192 148)(60 128 195 151)(63 131 198 154)(66 134 201 157)(69 110 204 160)(72 113 207 136)(75 116 210 139)(78 119 213 142)(81 122 216 145)
(1 33 97 169)(2 195 98 60)(3 129 99 152)(4 36 100 172)(5 198 101 63)(6 132 102 155)(7 39 103 175)(8 201 104 66)(9 135 105 158)(10 42 106 178)(11 204 107 69)(12 111 108 161)(13 45 82 181)(14 207 83 72)(15 114 84 137)(16 48 85 184)(17 210 86 75)(18 117 87 140)(19 51 88 187)(20 213 89 78)(21 120 90 143)(22 54 91 163)(23 216 92 81)(24 123 93 146)(25 30 94 166)(26 192 95 57)(27 126 96 149)(28 122 164 145)(29 55 165 190)(31 125 167 148)(32 58 168 193)(34 128 170 151)(35 61 171 196)(37 131 173 154)(38 64 174 199)(40 134 176 157)(41 67 177 202)(43 110 179 160)(44 70 180 205)(46 113 182 136)(47 73 183 208)(49 116 185 139)(50 76 186 211)(52 119 188 142)(53 79 189 214)(56 124 191 147)(59 127 194 150)(62 130 197 153)(65 133 200 156)(68 109 203 159)(71 112 206 162)(74 115 209 138)(77 118 212 141)(80 121 215 144)
(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)| (1,127,97,150)(2,34,98,170)(3,196,99,61)(4,130,100,153)(5,37,101,173)(6,199,102,64)(7,133,103,156)(8,40,104,176)(9,202,105,67)(10,109,106,159)(11,43,107,179)(12,205,108,70)(13,112,82,162)(14,46,83,182)(15,208,84,73)(16,115,85,138)(17,49,86,185)(18,211,87,76)(19,118,88,141)(20,52,89,188)(21,214,90,79)(22,121,91,144)(23,28,92,164)(24,190,93,55)(25,124,94,147)(26,31,95,167)(27,193,96,58)(29,123,165,146)(30,56,166,191)(32,126,168,149)(33,59,169,194)(35,129,171,152)(36,62,172,197)(38,132,174,155)(39,65,175,200)(41,135,177,158)(42,68,178,203)(44,111,180,161)(45,71,181,206)(47,114,183,137)(48,74,184,209)(50,117,186,140)(51,77,187,212)(53,120,189,143)(54,80,163,215)(57,125,192,148)(60,128,195,151)(63,131,198,154)(66,134,201,157)(69,110,204,160)(72,113,207,136)(75,116,210,139)(78,119,213,142)(81,122,216,145), (1,33,97,169)(2,195,98,60)(3,129,99,152)(4,36,100,172)(5,198,101,63)(6,132,102,155)(7,39,103,175)(8,201,104,66)(9,135,105,158)(10,42,106,178)(11,204,107,69)(12,111,108,161)(13,45,82,181)(14,207,83,72)(15,114,84,137)(16,48,85,184)(17,210,86,75)(18,117,87,140)(19,51,88,187)(20,213,89,78)(21,120,90,143)(22,54,91,163)(23,216,92,81)(24,123,93,146)(25,30,94,166)(26,192,95,57)(27,126,96,149)(28,122,164,145)(29,55,165,190)(31,125,167,148)(32,58,168,193)(34,128,170,151)(35,61,171,196)(37,131,173,154)(38,64,174,199)(40,134,176,157)(41,67,177,202)(43,110,179,160)(44,70,180,205)(46,113,182,136)(47,73,183,208)(49,116,185,139)(50,76,186,211)(52,119,188,142)(53,79,189,214)(56,124,191,147)(59,127,194,150)(62,130,197,153)(65,133,200,156)(68,109,203,159)(71,112,206,162)(74,115,209,138)(77,118,212,141)(80,121,215,144), (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( (1,127,97,150)(2,34,98,170)(3,196,99,61)(4,130,100,153)(5,37,101,173)(6,199,102,64)(7,133,103,156)(8,40,104,176)(9,202,105,67)(10,109,106,159)(11,43,107,179)(12,205,108,70)(13,112,82,162)(14,46,83,182)(15,208,84,73)(16,115,85,138)(17,49,86,185)(18,211,87,76)(19,118,88,141)(20,52,89,188)(21,214,90,79)(22,121,91,144)(23,28,92,164)(24,190,93,55)(25,124,94,147)(26,31,95,167)(27,193,96,58)(29,123,165,146)(30,56,166,191)(32,126,168,149)(33,59,169,194)(35,129,171,152)(36,62,172,197)(38,132,174,155)(39,65,175,200)(41,135,177,158)(42,68,178,203)(44,111,180,161)(45,71,181,206)(47,114,183,137)(48,74,184,209)(50,117,186,140)(51,77,187,212)(53,120,189,143)(54,80,163,215)(57,125,192,148)(60,128,195,151)(63,131,198,154)(66,134,201,157)(69,110,204,160)(72,113,207,136)(75,116,210,139)(78,119,213,142)(81,122,216,145), (1,33,97,169)(2,195,98,60)(3,129,99,152)(4,36,100,172)(5,198,101,63)(6,132,102,155)(7,39,103,175)(8,201,104,66)(9,135,105,158)(10,42,106,178)(11,204,107,69)(12,111,108,161)(13,45,82,181)(14,207,83,72)(15,114,84,137)(16,48,85,184)(17,210,86,75)(18,117,87,140)(19,51,88,187)(20,213,89,78)(21,120,90,143)(22,54,91,163)(23,216,92,81)(24,123,93,146)(25,30,94,166)(26,192,95,57)(27,126,96,149)(28,122,164,145)(29,55,165,190)(31,125,167,148)(32,58,168,193)(34,128,170,151)(35,61,171,196)(37,131,173,154)(38,64,174,199)(40,134,176,157)(41,67,177,202)(43,110,179,160)(44,70,180,205)(46,113,182,136)(47,73,183,208)(49,116,185,139)(50,76,186,211)(52,119,188,142)(53,79,189,214)(56,124,191,147)(59,127,194,150)(62,130,197,153)(65,133,200,156)(68,109,203,159)(71,112,206,162)(74,115,209,138)(77,118,212,141)(80,121,215,144), (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([(1,127,97,150),(2,34,98,170),(3,196,99,61),(4,130,100,153),(5,37,101,173),(6,199,102,64),(7,133,103,156),(8,40,104,176),(9,202,105,67),(10,109,106,159),(11,43,107,179),(12,205,108,70),(13,112,82,162),(14,46,83,182),(15,208,84,73),(16,115,85,138),(17,49,86,185),(18,211,87,76),(19,118,88,141),(20,52,89,188),(21,214,90,79),(22,121,91,144),(23,28,92,164),(24,190,93,55),(25,124,94,147),(26,31,95,167),(27,193,96,58),(29,123,165,146),(30,56,166,191),(32,126,168,149),(33,59,169,194),(35,129,171,152),(36,62,172,197),(38,132,174,155),(39,65,175,200),(41,135,177,158),(42,68,178,203),(44,111,180,161),(45,71,181,206),(47,114,183,137),(48,74,184,209),(50,117,186,140),(51,77,187,212),(53,120,189,143),(54,80,163,215),(57,125,192,148),(60,128,195,151),(63,131,198,154),(66,134,201,157),(69,110,204,160),(72,113,207,136),(75,116,210,139),(78,119,213,142),(81,122,216,145)], [(1,33,97,169),(2,195,98,60),(3,129,99,152),(4,36,100,172),(5,198,101,63),(6,132,102,155),(7,39,103,175),(8,201,104,66),(9,135,105,158),(10,42,106,178),(11,204,107,69),(12,111,108,161),(13,45,82,181),(14,207,83,72),(15,114,84,137),(16,48,85,184),(17,210,86,75),(18,117,87,140),(19,51,88,187),(20,213,89,78),(21,120,90,143),(22,54,91,163),(23,216,92,81),(24,123,93,146),(25,30,94,166),(26,192,95,57),(27,126,96,149),(28,122,164,145),(29,55,165,190),(31,125,167,148),(32,58,168,193),(34,128,170,151),(35,61,171,196),(37,131,173,154),(38,64,174,199),(40,134,176,157),(41,67,177,202),(43,110,179,160),(44,70,180,205),(46,113,182,136),(47,73,183,208),(49,116,185,139),(50,76,186,211),(52,119,188,142),(53,79,189,214),(56,124,191,147),(59,127,194,150),(62,130,197,153),(65,133,200,156),(68,109,203,159),(71,112,206,162),(74,115,209,138),(77,118,212,141),(80,121,215,144)], [(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 3A3B 4 6A6B9A···9F12A12B18A···18F27A···27R36A···36F54A···54R
order12334669···9121218···1827···2736···3654···54
size11116111···1661···14···46···64···4

63 irreducible representations

dim11112222333
type+-+
imageC1C3C9C27SL2(𝔽3)SL2(𝔽3)Q8⋊C9Q8⋊C27A4C3.A4C9.A4
kernelQ8⋊C27Q8×C9C3×Q8Q8C9C9C3C1C18C6C2
# reps1261812618126

Matrix representation of Q8⋊C27 in GL2(𝔽109) generated by

12
108108
,
3863
4871
,
7690
7894
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

Export

Subgroup lattice of Q8⋊C27 in TeX

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