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## G = Q8⋊3D27order 432 = 24·33

### The semidirect product of Q8 and D27 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C54 — Q8⋊3D27
 Chief series C1 — C3 — C9 — C27 — C54 — D54 — C4×D27 — Q8⋊3D27
 Lower central C27 — C54 — Q8⋊3D27
 Upper central C1 — C2 — Q8

Generators and relations for Q83D27
G = < a,b,c,d | a4=c27=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 704 in 80 conjugacy classes, 35 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, D4, Q8, C9, Dic3, C12, D6, C4○D4, D9, C18, C4×S3, D12, C3×Q8, C27, Dic9, C36, D18, Q83S3, D27, C54, C4×D9, D36, Q8×C9, Dic27, C108, D54, Q83D9, C4×D27, D108, Q8×C27, Q83D27
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, Q83S3, D27, C22×D9, D54, Q83D9, C22×D27, Q83D27

Smallest permutation representation of Q83D27
On 216 points
Generators in S216
```(1 107 48 69)(2 108 49 70)(3 82 50 71)(4 83 51 72)(5 84 52 73)(6 85 53 74)(7 86 54 75)(8 87 28 76)(9 88 29 77)(10 89 30 78)(11 90 31 79)(12 91 32 80)(13 92 33 81)(14 93 34 55)(15 94 35 56)(16 95 36 57)(17 96 37 58)(18 97 38 59)(19 98 39 60)(20 99 40 61)(21 100 41 62)(22 101 42 63)(23 102 43 64)(24 103 44 65)(25 104 45 66)(26 105 46 67)(27 106 47 68)(109 182 157 191)(110 183 158 192)(111 184 159 193)(112 185 160 194)(113 186 161 195)(114 187 162 196)(115 188 136 197)(116 189 137 198)(117 163 138 199)(118 164 139 200)(119 165 140 201)(120 166 141 202)(121 167 142 203)(122 168 143 204)(123 169 144 205)(124 170 145 206)(125 171 146 207)(126 172 147 208)(127 173 148 209)(128 174 149 210)(129 175 150 211)(130 176 151 212)(131 177 152 213)(132 178 153 214)(133 179 154 215)(134 180 155 216)(135 181 156 190)
(1 136 48 115)(2 137 49 116)(3 138 50 117)(4 139 51 118)(5 140 52 119)(6 141 53 120)(7 142 54 121)(8 143 28 122)(9 144 29 123)(10 145 30 124)(11 146 31 125)(12 147 32 126)(13 148 33 127)(14 149 34 128)(15 150 35 129)(16 151 36 130)(17 152 37 131)(18 153 38 132)(19 154 39 133)(20 155 40 134)(21 156 41 135)(22 157 42 109)(23 158 43 110)(24 159 44 111)(25 160 45 112)(26 161 46 113)(27 162 47 114)(55 210 93 174)(56 211 94 175)(57 212 95 176)(58 213 96 177)(59 214 97 178)(60 215 98 179)(61 216 99 180)(62 190 100 181)(63 191 101 182)(64 192 102 183)(65 193 103 184)(66 194 104 185)(67 195 105 186)(68 196 106 187)(69 197 107 188)(70 198 108 189)(71 199 82 163)(72 200 83 164)(73 201 84 165)(74 202 85 166)(75 203 86 167)(76 204 87 168)(77 205 88 169)(78 206 89 170)(79 207 90 171)(80 208 91 172)(81 209 92 173)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 81)(17 80)(18 79)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(25 72)(26 71)(27 70)(28 100)(29 99)(30 98)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(109 167)(110 166)(111 165)(112 164)(113 163)(114 189)(115 188)(116 187)(117 186)(118 185)(119 184)(120 183)(121 182)(122 181)(123 180)(124 179)(125 178)(126 177)(127 176)(128 175)(129 174)(130 173)(131 172)(132 171)(133 170)(134 169)(135 168)(136 197)(137 196)(138 195)(139 194)(140 193)(141 192)(142 191)(143 190)(144 216)(145 215)(146 214)(147 213)(148 212)(149 211)(150 210)(151 209)(152 208)(153 207)(154 206)(155 205)(156 204)(157 203)(158 202)(159 201)(160 200)(161 199)(162 198)```

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

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

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

75 conjugacy classes

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

75 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D6 C4○D4 D9 D18 D27 D54 Q8⋊3S3 Q8⋊3D9 Q8⋊3D27 kernel Q8⋊3D27 C4×D27 D108 Q8×C27 Q8×C9 C36 C27 C3×Q8 C12 Q8 C4 C9 C3 C1 # reps 1 3 3 1 1 3 2 3 9 9 27 1 3 9

Matrix representation of Q83D27 in GL4(𝔽109) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 108 0
,
 108 0 0 0 0 108 0 0 0 0 0 33 0 0 33 0
,
 30 93 0 0 16 46 0 0 0 0 1 0 0 0 0 1
,
 77 27 0 0 59 32 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,0,108,0,0,1,0],[108,0,0,0,0,108,0,0,0,0,0,33,0,0,33,0],[30,16,0,0,93,46,0,0,0,0,1,0,0,0,0,1],[77,59,0,0,27,32,0,0,0,0,0,1,0,0,1,0] >;`

Q83D27 in GAP, Magma, Sage, TeX

`Q_8\rtimes_3D_{27}`
`% in TeX`

`G:=Group("Q8:3D27");`
`// GroupNames label`

`G:=SmallGroup(432,50);`
`// by ID`

`G=gap.SmallGroup(432,50);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,2804,557,10085,292,14118]);`
`// 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,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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