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## G = Q8⋊D27order 432 = 24·33

### The semidirect product of Q8 and D27 acting via D27/C9=S3

Aliases: Q8⋊D27, C18.2S4, C9.GL2(𝔽3), Q8⋊C27⋊C2, C3.(Q8⋊D9), (Q8×C9).2S3, (C3×Q8).2D9, C6.2(C3.S4), C2.3(C9.S4), SmallGroup(432,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8⋊C27 — Q8⋊D27
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C9 — Q8⋊C27 — Q8⋊D27
 Lower central Q8⋊C27 — Q8⋊D27
 Upper central C1 — C2

Generators and relations for Q8⋊D27
G = < a,b,c,d | a4=c27=d2=1, b2=a2, bab-1=dbd=a-1, cac-1=b, dad=a2b, cbc-1=ab, dcd=c-1 >

108C2
3C4
54C22
36S3
27D4
27C8
3C12
18D6
12D9
4C27
27SD16
9D12
3C36
6D18
4D27
4C54
4D27
3D36
4D54

Smallest permutation representation of Q8⋊D27
On 216 points
Generators in S216
```(1 28 205 78)(2 182 206 94)(3 127 207 162)(4 31 208 81)(5 185 209 97)(6 130 210 138)(7 34 211 57)(8 188 212 100)(9 133 213 141)(10 37 214 60)(11 164 215 103)(12 109 216 144)(13 40 190 63)(14 167 191 106)(15 112 192 147)(16 43 193 66)(17 170 194 82)(18 115 195 150)(19 46 196 69)(20 173 197 85)(21 118 198 153)(22 49 199 72)(23 176 200 88)(24 121 201 156)(25 52 202 75)(26 179 203 91)(27 124 204 159)(29 126 79 161)(30 95 80 183)(32 129 55 137)(33 98 56 186)(35 132 58 140)(36 101 59 189)(38 135 61 143)(39 104 62 165)(41 111 64 146)(42 107 65 168)(44 114 67 149)(45 83 68 171)(47 117 70 152)(48 86 71 174)(50 120 73 155)(51 89 74 177)(53 123 76 158)(54 92 77 180)(84 116 172 151)(87 119 175 154)(90 122 178 157)(93 125 181 160)(96 128 184 136)(99 131 187 139)(102 134 163 142)(105 110 166 145)(108 113 169 148)
(1 181 205 93)(2 126 206 161)(3 30 207 80)(4 184 208 96)(5 129 209 137)(6 33 210 56)(7 187 211 99)(8 132 212 140)(9 36 213 59)(10 163 214 102)(11 135 215 143)(12 39 216 62)(13 166 190 105)(14 111 191 146)(15 42 192 65)(16 169 193 108)(17 114 194 149)(18 45 195 68)(19 172 196 84)(20 117 197 152)(21 48 198 71)(22 175 199 87)(23 120 200 155)(24 51 201 74)(25 178 202 90)(26 123 203 158)(27 54 204 77)(28 125 78 160)(29 94 79 182)(31 128 81 136)(32 97 55 185)(34 131 57 139)(35 100 58 188)(37 134 60 142)(38 103 61 164)(40 110 63 145)(41 106 64 167)(43 113 66 148)(44 82 67 170)(46 116 69 151)(47 85 70 173)(49 119 72 154)(50 88 73 176)(52 122 75 157)(53 91 76 179)(83 115 171 150)(86 118 174 153)(89 121 177 156)(92 124 180 159)(95 127 183 162)(98 130 186 138)(101 133 189 141)(104 109 165 144)(107 112 168 147)
(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)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 93)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 84)(38 83)(39 82)(40 108)(41 107)(42 106)(43 105)(44 104)(45 103)(46 102)(47 101)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 177)(56 176)(57 175)(58 174)(59 173)(60 172)(61 171)(62 170)(63 169)(64 168)(65 167)(66 166)(67 165)(68 164)(69 163)(70 189)(71 188)(72 187)(73 186)(74 185)(75 184)(76 183)(77 182)(78 181)(79 180)(80 179)(81 178)(109 149)(110 148)(111 147)(112 146)(113 145)(114 144)(115 143)(116 142)(117 141)(118 140)(119 139)(120 138)(121 137)(122 136)(123 162)(124 161)(125 160)(126 159)(127 158)(128 157)(129 156)(130 155)(131 154)(132 153)(133 152)(134 151)(135 150)(190 193)(191 192)(194 216)(195 215)(196 214)(197 213)(198 212)(199 211)(200 210)(201 209)(202 208)(203 207)(204 206)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 3 4 6 8A 8B 9A 9B 9C 12 18A 18B 18C 27A ··· 27I 36A 36B 36C 54A ··· 54I order 1 2 2 3 4 6 8 8 9 9 9 12 18 18 18 27 ··· 27 36 36 36 54 ··· 54 size 1 1 108 2 6 2 54 54 2 2 2 12 2 2 2 8 ··· 8 12 12 12 8 ··· 8

36 irreducible representations

 dim 1 1 2 2 2 2 3 4 4 4 6 6 type + + + + + + + + + + + image C1 C2 S3 D9 GL2(𝔽3) D27 S4 GL2(𝔽3) Q8⋊D9 Q8⋊D27 C3.S4 C9.S4 kernel Q8⋊D27 Q8⋊C27 Q8×C9 C3×Q8 C9 Q8 C18 C9 C3 C1 C6 C2 # reps 1 1 1 3 2 9 2 1 3 9 1 3

Matrix representation of Q8⋊D27 in GL4(𝔽433) generated by

 1 0 0 0 0 1 0 0 0 0 1 431 0 0 1 432
,
 1 0 0 0 0 1 0 0 0 0 70 431 0 0 69 363
,
 401 130 0 0 303 271 0 0 0 0 363 71 0 0 182 69
,
 1 0 0 0 432 432 0 0 0 0 1 0 0 0 252 432
`G:=sub<GL(4,GF(433))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,431,432],[1,0,0,0,0,1,0,0,0,0,70,69,0,0,431,363],[401,303,0,0,130,271,0,0,0,0,363,182,0,0,71,69],[1,432,0,0,0,432,0,0,0,0,1,252,0,0,0,432] >;`

Q8⋊D27 in GAP, Magma, Sage, TeX

`Q_8\rtimes D_{27}`
`% in TeX`

`G:=Group("Q8:D27");`
`// GroupNames label`

`G:=SmallGroup(432,38);`
`// by ID`

`G=gap.SmallGroup(432,38);`
`# by ID`

`G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,141,218,632,142,1011,3784,5681,172,2273,3414,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^27=d^2=1,b^2=a^2,b*a*b^-1=d*b*d=a^-1,c*a*c^-1=b,d*a*d=a^2*b,c*b*c^-1=a*b,d*c*d=c^-1>;`
`// generators/relations`

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