Copied to
clipboard

G = Q8×C27order 216 = 23·33

Direct product of C27 and Q8

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

C1C2 — Q8×C27
C1C3C9C18C54C108 — Q8×C27
C1C2 — Q8×C27
C1C54 — Q8×C27

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

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

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

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

Q8×C27 is a maximal subgroup of   C27⋊Q16  Q82D27  Q83D27

135 conjugacy classes

class 1  2 3A3B4A4B4C6A6B9A···9F12A···12F18A···18F27A···27R36A···36R54A···54R108A···108BB
order1233444669···912···1218···1827···2736···3654···54108···108
size1111222111···12···21···11···12···21···12···2

135 irreducible representations

dim111111112222
type++-
imageC1C2C3C6C9C18C27C54Q8C3×Q8Q8×C9Q8×C27
kernelQ8×C27C108Q8×C9C36C3×Q8C12Q8C4C27C9C3C1
# reps1326618185412618

Matrix representation of Q8×C27 in GL3(𝔽109) generated by

9700
0160
0016
,
10800
001
01080
,
100
06757
05742
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

Export

Subgroup lattice of Q8×C27 in TeX

׿
×
𝔽