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## G = Q8⋊2D27order 432 = 24·33

### The semidirect product of Q8 and D27 acting via D27/C27=C2

Aliases: Q82D27, C36.4D6, C4.4D54, C273SD16, C12.4D18, C54.10D4, D108.2C2, C108.4C22, C27⋊C83C2, (Q8×C27)⋊1C2, (Q8×C9).6S3, (C3×Q8).6D9, C3.(Q82D9), C9.(Q82S3), C6.19(C9⋊D4), C2.7(C27⋊D4), C18.19(C3⋊D4), SmallGroup(432,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C108 — Q8⋊2D27
 Chief series C1 — C3 — C9 — C27 — C54 — C108 — D108 — Q8⋊2D27
 Lower central C27 — C54 — C108 — Q8⋊2D27
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q82D27
G = < a,b,c,d | a4=c27=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

Smallest permutation representation of Q82D27
On 216 points
Generators in S216
```(1 99 37 57)(2 100 38 58)(3 101 39 59)(4 102 40 60)(5 103 41 61)(6 104 42 62)(7 105 43 63)(8 106 44 64)(9 107 45 65)(10 108 46 66)(11 82 47 67)(12 83 48 68)(13 84 49 69)(14 85 50 70)(15 86 51 71)(16 87 52 72)(17 88 53 73)(18 89 54 74)(19 90 28 75)(20 91 29 76)(21 92 30 77)(22 93 31 78)(23 94 32 79)(24 95 33 80)(25 96 34 81)(26 97 35 55)(27 98 36 56)(109 183 145 207)(110 184 146 208)(111 185 147 209)(112 186 148 210)(113 187 149 211)(114 188 150 212)(115 189 151 213)(116 163 152 214)(117 164 153 215)(118 165 154 216)(119 166 155 190)(120 167 156 191)(121 168 157 192)(122 169 158 193)(123 170 159 194)(124 171 160 195)(125 172 161 196)(126 173 162 197)(127 174 136 198)(128 175 137 199)(129 176 138 200)(130 177 139 201)(131 178 140 202)(132 179 141 203)(133 180 142 204)(134 181 143 205)(135 182 144 206)
(1 143 37 134)(2 144 38 135)(3 145 39 109)(4 146 40 110)(5 147 41 111)(6 148 42 112)(7 149 43 113)(8 150 44 114)(9 151 45 115)(10 152 46 116)(11 153 47 117)(12 154 48 118)(13 155 49 119)(14 156 50 120)(15 157 51 121)(16 158 52 122)(17 159 53 123)(18 160 54 124)(19 161 28 125)(20 162 29 126)(21 136 30 127)(22 137 31 128)(23 138 32 129)(24 139 33 130)(25 140 34 131)(26 141 35 132)(27 142 36 133)(55 203 97 179)(56 204 98 180)(57 205 99 181)(58 206 100 182)(59 207 101 183)(60 208 102 184)(61 209 103 185)(62 210 104 186)(63 211 105 187)(64 212 106 188)(65 213 107 189)(66 214 108 163)(67 215 82 164)(68 216 83 165)(69 190 84 166)(70 191 85 167)(71 192 86 168)(72 193 87 169)(73 194 88 170)(74 195 89 171)(75 196 90 172)(76 197 91 173)(77 198 92 174)(78 199 93 175)(79 200 94 176)(80 201 95 177)(81 202 96 178)
(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 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(47 54)(48 53)(49 52)(50 51)(55 101)(56 100)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(109 179)(110 178)(111 177)(112 176)(113 175)(114 174)(115 173)(116 172)(117 171)(118 170)(119 169)(120 168)(121 167)(122 166)(123 165)(124 164)(125 163)(126 189)(127 188)(128 187)(129 186)(130 185)(131 184)(132 183)(133 182)(134 181)(135 180)(136 212)(137 211)(138 210)(139 209)(140 208)(141 207)(142 206)(143 205)(144 204)(145 203)(146 202)(147 201)(148 200)(149 199)(150 198)(151 197)(152 196)(153 195)(154 194)(155 193)(156 192)(157 191)(158 190)(159 216)(160 215)(161 214)(162 213)```

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 SD16 D9 C3⋊D4 D18 D27 C9⋊D4 D54 C27⋊D4 Q8⋊2S3 Q8⋊2D9 Q8⋊2D27 kernel Q8⋊2D27 C27⋊C8 D108 Q8×C27 Q8×C9 C54 C36 C27 C3×Q8 C18 C12 Q8 C6 C4 C2 C9 C3 C1 # reps 1 1 1 1 1 1 1 2 3 2 3 9 6 9 18 1 3 9

Matrix representation of Q82D27 in GL4(𝔽433) generated by

 1 0 0 0 0 1 0 0 0 0 432 431 0 0 1 1
,
 1 0 0 0 0 1 0 0 0 0 0 69 0 0 251 0
,
 288 182 0 0 251 106 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 432 432 0 0 0 0 1 0 0 0 432 432
`G:=sub<GL(4,GF(433))| [1,0,0,0,0,1,0,0,0,0,432,1,0,0,431,1],[1,0,0,0,0,1,0,0,0,0,0,251,0,0,69,0],[288,251,0,0,182,106,0,0,0,0,1,0,0,0,0,1],[1,432,0,0,0,432,0,0,0,0,1,432,0,0,0,432] >;`

Q82D27 in GAP, Magma, Sage, TeX

`Q_8\rtimes_2D_{27}`
`% in TeX`

`G:=Group("Q8:2D27");`
`// GroupNames label`

`G:=SmallGroup(432,18);`
`// by ID`

`G=gap.SmallGroup(432,18);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,2804,557,10085,292,14118]);`
`// Polycyclic`

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

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