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## G = C54.D4order 432 = 24·33

### 7th non-split extension by C54 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C54 — C54.D4
 Chief series C1 — C3 — C9 — C27 — C54 — C2×C54 — C2×Dic27 — C54.D4
 Lower central C27 — C54 — C54.D4
 Upper central C1 — C22 — C23

Generators and relations for C54.D4
G = < a,b,c | a54=b4=1, c2=a27, bab-1=cac-1=a-1, cbc-1=a27b-1 >

Subgroups: 344 in 68 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, Dic3, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C18, C2×Dic3, C22×C6, C27, Dic9, C2×C18, C2×C18, C2×C18, C6.D4, C54, C54, C54, C2×Dic9, C22×C18, Dic27, C2×C54, C2×C54, C2×C54, C18.D4, C2×Dic27, C22×C54, C54.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D9, C2×Dic3, C3⋊D4, Dic9, D18, C6.D4, D27, C2×Dic9, C9⋊D4, Dic27, D54, C18.D4, C2×Dic27, C27⋊D4, C54.D4

Smallest permutation representation of C54.D4
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)
(1 168 121 55)(2 167 122 108)(3 166 123 107)(4 165 124 106)(5 164 125 105)(6 163 126 104)(7 216 127 103)(8 215 128 102)(9 214 129 101)(10 213 130 100)(11 212 131 99)(12 211 132 98)(13 210 133 97)(14 209 134 96)(15 208 135 95)(16 207 136 94)(17 206 137 93)(18 205 138 92)(19 204 139 91)(20 203 140 90)(21 202 141 89)(22 201 142 88)(23 200 143 87)(24 199 144 86)(25 198 145 85)(26 197 146 84)(27 196 147 83)(28 195 148 82)(29 194 149 81)(30 193 150 80)(31 192 151 79)(32 191 152 78)(33 190 153 77)(34 189 154 76)(35 188 155 75)(36 187 156 74)(37 186 157 73)(38 185 158 72)(39 184 159 71)(40 183 160 70)(41 182 161 69)(42 181 162 68)(43 180 109 67)(44 179 110 66)(45 178 111 65)(46 177 112 64)(47 176 113 63)(48 175 114 62)(49 174 115 61)(50 173 116 60)(51 172 117 59)(52 171 118 58)(53 170 119 57)(54 169 120 56)
(1 82 28 55)(2 81 29 108)(3 80 30 107)(4 79 31 106)(5 78 32 105)(6 77 33 104)(7 76 34 103)(8 75 35 102)(9 74 36 101)(10 73 37 100)(11 72 38 99)(12 71 39 98)(13 70 40 97)(14 69 41 96)(15 68 42 95)(16 67 43 94)(17 66 44 93)(18 65 45 92)(19 64 46 91)(20 63 47 90)(21 62 48 89)(22 61 49 88)(23 60 50 87)(24 59 51 86)(25 58 52 85)(26 57 53 84)(27 56 54 83)(109 207 136 180)(110 206 137 179)(111 205 138 178)(112 204 139 177)(113 203 140 176)(114 202 141 175)(115 201 142 174)(116 200 143 173)(117 199 144 172)(118 198 145 171)(119 197 146 170)(120 196 147 169)(121 195 148 168)(122 194 149 167)(123 193 150 166)(124 192 151 165)(125 191 152 164)(126 190 153 163)(127 189 154 216)(128 188 155 215)(129 187 156 214)(130 186 157 213)(131 185 158 212)(132 184 159 211)(133 183 160 210)(134 182 161 209)(135 181 162 208)```

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

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

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

114 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A ··· 6G 9A 9B 9C 18A ··· 18U 27A ··· 27I 54A ··· 54BK order 1 2 2 2 2 2 3 4 4 4 4 6 ··· 6 9 9 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 1 1 2 2 2 54 54 54 54 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - + + - + image C1 C2 C2 C4 S3 D4 Dic3 D6 D9 C3⋊D4 Dic9 D18 D27 C9⋊D4 Dic27 D54 C27⋊D4 kernel C54.D4 C2×Dic27 C22×C54 C2×C54 C22×C18 C54 C2×C18 C2×C18 C22×C6 C18 C2×C6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 1 4 1 2 2 1 3 4 6 3 9 12 18 9 36

Matrix representation of C54.D4 in GL4(𝔽109) generated by

 43 0 0 0 26 71 0 0 0 0 80 0 0 0 0 15
,
 39 42 0 0 78 70 0 0 0 0 0 1 0 0 108 0
,
 39 42 0 0 52 70 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(109))| [43,26,0,0,0,71,0,0,0,0,80,0,0,0,0,15],[39,78,0,0,42,70,0,0,0,0,0,108,0,0,1,0],[39,52,0,0,42,70,0,0,0,0,0,1,0,0,1,0] >;`

C54.D4 in GAP, Magma, Sage, TeX

`C_{54}.D_4`
`% in TeX`

`G:=Group("C54.D4");`
`// GroupNames label`

`G:=SmallGroup(432,19);`
`// by ID`

`G=gap.SmallGroup(432,19);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,2804,557,10085,292,14118]);`
`// Polycyclic`

`G:=Group<a,b,c|a^54=b^4=1,c^2=a^27,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^27*b^-1>;`
`// generators/relations`

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