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G = Q8.C54order 432 = 24·33

The non-split extension by Q8 of C54 acting via C54/C18=C3

non-abelian, soluble

Aliases: Q8.C54, C36.3A4, C4○D4⋊C27, Q8⋊C272C2, 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

C1C2Q8 — Q8.C54
C1C2Q8C3×Q8Q8×C9Q8⋊C27 — Q8.C54
Q8 — Q8.C54
C1C36

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 >

6C2
3C4
3C22
6C6
3C2×C4
3D4
3C12
3C2×C6
6C18
4C27
3C3×D4
3C2×C12
3C36
3C2×C18
4C54
3D4×C9
3C2×C36
4C108

Smallest permutation representation of Q8.C54
On 216 points
Generators in S216
(1 153 55 207)(2 127 56 181)(3 30 57 84)(4 156 58 210)(5 130 59 184)(6 33 60 87)(7 159 61 213)(8 133 62 187)(9 36 63 90)(10 162 64 216)(11 136 65 190)(12 39 66 93)(13 165 67 111)(14 139 68 193)(15 42 69 96)(16 168 70 114)(17 142 71 196)(18 45 72 99)(19 171 73 117)(20 145 74 199)(21 48 75 102)(22 174 76 120)(23 148 77 202)(24 51 78 105)(25 177 79 123)(26 151 80 205)(27 54 81 108)(28 180 82 126)(29 154 83 208)(31 183 85 129)(32 157 86 211)(34 186 88 132)(35 160 89 214)(37 189 91 135)(38 163 92 109)(40 192 94 138)(41 166 95 112)(43 195 97 141)(44 169 98 115)(46 198 100 144)(47 172 101 118)(49 201 103 147)(50 175 104 121)(52 204 106 150)(53 178 107 124)(110 191 164 137)(113 194 167 140)(116 197 170 143)(119 200 173 146)(122 203 176 149)(125 206 179 152)(128 209 182 155)(131 212 185 158)(134 215 188 161)
(1 28 55 82)(2 154 56 208)(3 128 57 182)(4 31 58 85)(5 157 59 211)(6 131 60 185)(7 34 61 88)(8 160 62 214)(9 134 63 188)(10 37 64 91)(11 163 65 109)(12 137 66 191)(13 40 67 94)(14 166 68 112)(15 140 69 194)(16 43 70 97)(17 169 71 115)(18 143 72 197)(19 46 73 100)(20 172 74 118)(21 146 75 200)(22 49 76 103)(23 175 77 121)(24 149 78 203)(25 52 79 106)(26 178 80 124)(27 152 81 206)(29 181 83 127)(30 155 84 209)(32 184 86 130)(33 158 87 212)(35 187 89 133)(36 161 90 215)(38 190 92 136)(39 164 93 110)(41 193 95 139)(42 167 96 113)(44 196 98 142)(45 170 99 116)(47 199 101 145)(48 173 102 119)(50 202 104 148)(51 176 105 122)(53 205 107 151)(54 179 108 125)(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 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)| (1,153,55,207)(2,127,56,181)(3,30,57,84)(4,156,58,210)(5,130,59,184)(6,33,60,87)(7,159,61,213)(8,133,62,187)(9,36,63,90)(10,162,64,216)(11,136,65,190)(12,39,66,93)(13,165,67,111)(14,139,68,193)(15,42,69,96)(16,168,70,114)(17,142,71,196)(18,45,72,99)(19,171,73,117)(20,145,74,199)(21,48,75,102)(22,174,76,120)(23,148,77,202)(24,51,78,105)(25,177,79,123)(26,151,80,205)(27,54,81,108)(28,180,82,126)(29,154,83,208)(31,183,85,129)(32,157,86,211)(34,186,88,132)(35,160,89,214)(37,189,91,135)(38,163,92,109)(40,192,94,138)(41,166,95,112)(43,195,97,141)(44,169,98,115)(46,198,100,144)(47,172,101,118)(49,201,103,147)(50,175,104,121)(52,204,106,150)(53,178,107,124)(110,191,164,137)(113,194,167,140)(116,197,170,143)(119,200,173,146)(122,203,176,149)(125,206,179,152)(128,209,182,155)(131,212,185,158)(134,215,188,161), (1,28,55,82)(2,154,56,208)(3,128,57,182)(4,31,58,85)(5,157,59,211)(6,131,60,185)(7,34,61,88)(8,160,62,214)(9,134,63,188)(10,37,64,91)(11,163,65,109)(12,137,66,191)(13,40,67,94)(14,166,68,112)(15,140,69,194)(16,43,70,97)(17,169,71,115)(18,143,72,197)(19,46,73,100)(20,172,74,118)(21,146,75,200)(22,49,76,103)(23,175,77,121)(24,149,78,203)(25,52,79,106)(26,178,80,124)(27,152,81,206)(29,181,83,127)(30,155,84,209)(32,184,86,130)(33,158,87,212)(35,187,89,133)(36,161,90,215)(38,190,92,136)(39,164,93,110)(41,193,95,139)(42,167,96,113)(44,196,98,142)(45,170,99,116)(47,199,101,145)(48,173,102,119)(50,202,104,148)(51,176,105,122)(53,205,107,151)(54,179,108,125)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(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( (1,153,55,207)(2,127,56,181)(3,30,57,84)(4,156,58,210)(5,130,59,184)(6,33,60,87)(7,159,61,213)(8,133,62,187)(9,36,63,90)(10,162,64,216)(11,136,65,190)(12,39,66,93)(13,165,67,111)(14,139,68,193)(15,42,69,96)(16,168,70,114)(17,142,71,196)(18,45,72,99)(19,171,73,117)(20,145,74,199)(21,48,75,102)(22,174,76,120)(23,148,77,202)(24,51,78,105)(25,177,79,123)(26,151,80,205)(27,54,81,108)(28,180,82,126)(29,154,83,208)(31,183,85,129)(32,157,86,211)(34,186,88,132)(35,160,89,214)(37,189,91,135)(38,163,92,109)(40,192,94,138)(41,166,95,112)(43,195,97,141)(44,169,98,115)(46,198,100,144)(47,172,101,118)(49,201,103,147)(50,175,104,121)(52,204,106,150)(53,178,107,124)(110,191,164,137)(113,194,167,140)(116,197,170,143)(119,200,173,146)(122,203,176,149)(125,206,179,152)(128,209,182,155)(131,212,185,158)(134,215,188,161), (1,28,55,82)(2,154,56,208)(3,128,57,182)(4,31,58,85)(5,157,59,211)(6,131,60,185)(7,34,61,88)(8,160,62,214)(9,134,63,188)(10,37,64,91)(11,163,65,109)(12,137,66,191)(13,40,67,94)(14,166,68,112)(15,140,69,194)(16,43,70,97)(17,169,71,115)(18,143,72,197)(19,46,73,100)(20,172,74,118)(21,146,75,200)(22,49,76,103)(23,175,77,121)(24,149,78,203)(25,52,79,106)(26,178,80,124)(27,152,81,206)(29,181,83,127)(30,155,84,209)(32,184,86,130)(33,158,87,212)(35,187,89,133)(36,161,90,215)(38,190,92,136)(39,164,93,110)(41,193,95,139)(42,167,96,113)(44,196,98,142)(45,170,99,116)(47,199,101,145)(48,173,102,119)(50,202,104,148)(51,176,105,122)(53,205,107,151)(54,179,108,125)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(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([(1,153,55,207),(2,127,56,181),(3,30,57,84),(4,156,58,210),(5,130,59,184),(6,33,60,87),(7,159,61,213),(8,133,62,187),(9,36,63,90),(10,162,64,216),(11,136,65,190),(12,39,66,93),(13,165,67,111),(14,139,68,193),(15,42,69,96),(16,168,70,114),(17,142,71,196),(18,45,72,99),(19,171,73,117),(20,145,74,199),(21,48,75,102),(22,174,76,120),(23,148,77,202),(24,51,78,105),(25,177,79,123),(26,151,80,205),(27,54,81,108),(28,180,82,126),(29,154,83,208),(31,183,85,129),(32,157,86,211),(34,186,88,132),(35,160,89,214),(37,189,91,135),(38,163,92,109),(40,192,94,138),(41,166,95,112),(43,195,97,141),(44,169,98,115),(46,198,100,144),(47,172,101,118),(49,201,103,147),(50,175,104,121),(52,204,106,150),(53,178,107,124),(110,191,164,137),(113,194,167,140),(116,197,170,143),(119,200,173,146),(122,203,176,149),(125,206,179,152),(128,209,182,155),(131,212,185,158),(134,215,188,161)], [(1,28,55,82),(2,154,56,208),(3,128,57,182),(4,31,58,85),(5,157,59,211),(6,131,60,185),(7,34,61,88),(8,160,62,214),(9,134,63,188),(10,37,64,91),(11,163,65,109),(12,137,66,191),(13,40,67,94),(14,166,68,112),(15,140,69,194),(16,43,70,97),(17,169,71,115),(18,143,72,197),(19,46,73,100),(20,172,74,118),(21,146,75,200),(22,49,76,103),(23,175,77,121),(24,149,78,203),(25,52,79,106),(26,178,80,124),(27,152,81,206),(29,181,83,127),(30,155,84,209),(32,184,86,130),(33,158,87,212),(35,187,89,133),(36,161,90,215),(38,190,92,136),(39,164,93,110),(41,193,95,139),(42,167,96,113),(44,196,98,142),(45,170,99,116),(47,199,101,145),(48,173,102,119),(50,202,104,148),(51,176,105,122),(53,205,107,151),(54,179,108,125),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(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 2A2B3A3B4A4B4C6A6B6C6D9A···9F12A12B12C12D12E12F18A···18F18G···18L27A···27R36A···36L36M···36R54A···54R108A···108AJ
order1223344466669···912121212121218···1818···1827···2736···3636···3654···54108···108
size1161111611661···11111661···16···64···41···16···64···44···4

126 irreducible representations

dim11111111222333333
type++++
imageC1C2C3C6C9C18C27C54C4.A4Q8.C18Q8.C54A4C2×A4C3.A4C2×C3.A4C9.A4C2×C9.A4
kernelQ8.C54Q8⋊C27C9×C4○D4Q8×C9C3×C4○D4C3×Q8C4○D4Q8C9C3C1C36C18C12C6C4C2
# reps112266181861236112266

Matrix representation of Q8.C54 in GL2(𝔽109) generated by

0108
10
,
760
033
,
2857
8157
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

Subgroup lattice of Q8.C54 in TeX

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