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## G = C24⋊C27order 432 = 24·33

### 2nd semidirect product of C24 and C27 acting via C27/C9=C3

Aliases: C242C27, (C2×C18).4A4, C9.(C22⋊A4), C22⋊(C9.A4), C3.(C24⋊C9), (C23×C6).2C9, (C23×C18).2C3, (C2×C6).3(C3.A4), SmallGroup(432,226)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C27
 Chief series C1 — C22 — C24 — C23×C6 — C23×C18 — C24⋊C27
 Lower central C24 — C24⋊C27
 Upper central C1 — C9

Generators and relations for C24⋊C27
G = < a,b,c,d,e | a2=b2=c2=d2=e27=1, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, ebe-1=a, ece-1=cd=dc, ede-1=c >

Subgroups: 238 in 88 conjugacy classes, 22 normal (7 characteristic)
C1, C2, C3, C22, C22, C6, C23, C9, C2×C6, C2×C6, C24, C18, C22×C6, C27, C2×C18, C2×C18, C23×C6, C22×C18, C9.A4, C23×C18, C24⋊C27
Quotients: C1, C3, C9, A4, C27, C3.A4, C22⋊A4, C9.A4, C24⋊C9, C24⋊C27

Smallest permutation representation of C24⋊C27
On 108 points
Generators in S108
```(1 106)(2 42)(3 80)(4 82)(5 45)(6 56)(7 85)(8 48)(9 59)(10 88)(11 51)(12 62)(13 91)(14 54)(15 65)(16 94)(17 30)(18 68)(19 97)(20 33)(21 71)(22 100)(23 36)(24 74)(25 103)(26 39)(27 77)(28 93)(29 66)(31 96)(32 69)(34 99)(35 72)(37 102)(38 75)(40 105)(41 78)(43 108)(44 81)(46 84)(47 57)(49 87)(50 60)(52 90)(53 63)(55 83)(58 86)(61 89)(64 92)(67 95)(70 98)(73 101)(76 104)(79 107)
(1 78)(2 107)(3 43)(4 81)(5 83)(6 46)(7 57)(8 86)(9 49)(10 60)(11 89)(12 52)(13 63)(14 92)(15 28)(16 66)(17 95)(18 31)(19 69)(20 98)(21 34)(22 72)(23 101)(24 37)(25 75)(26 104)(27 40)(29 94)(30 67)(32 97)(33 70)(35 100)(36 73)(38 103)(39 76)(41 106)(42 79)(44 82)(45 55)(47 85)(48 58)(50 88)(51 61)(53 91)(54 64)(56 84)(59 87)(62 90)(65 93)(68 96)(71 99)(74 102)(77 105)(80 108)
(1 106)(3 108)(4 82)(6 84)(7 85)(9 87)(10 88)(12 90)(13 91)(15 93)(16 94)(18 96)(19 97)(21 99)(22 100)(24 102)(25 103)(27 105)(28 65)(29 66)(31 68)(32 69)(34 71)(35 72)(37 74)(38 75)(40 77)(41 78)(43 80)(44 81)(46 56)(47 57)(49 59)(50 60)(52 62)(53 63)
(1 106)(2 107)(4 82)(5 83)(7 85)(8 86)(10 88)(11 89)(13 91)(14 92)(16 94)(17 95)(19 97)(20 98)(22 100)(23 101)(25 103)(26 104)(29 66)(30 67)(32 69)(33 70)(35 72)(36 73)(38 75)(39 76)(41 78)(42 79)(44 81)(45 55)(47 57)(48 58)(50 60)(51 61)(53 63)(54 64)
(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)```

`G:=sub<Sym(108)| (1,106)(2,42)(3,80)(4,82)(5,45)(6,56)(7,85)(8,48)(9,59)(10,88)(11,51)(12,62)(13,91)(14,54)(15,65)(16,94)(17,30)(18,68)(19,97)(20,33)(21,71)(22,100)(23,36)(24,74)(25,103)(26,39)(27,77)(28,93)(29,66)(31,96)(32,69)(34,99)(35,72)(37,102)(38,75)(40,105)(41,78)(43,108)(44,81)(46,84)(47,57)(49,87)(50,60)(52,90)(53,63)(55,83)(58,86)(61,89)(64,92)(67,95)(70,98)(73,101)(76,104)(79,107), (1,78)(2,107)(3,43)(4,81)(5,83)(6,46)(7,57)(8,86)(9,49)(10,60)(11,89)(12,52)(13,63)(14,92)(15,28)(16,66)(17,95)(18,31)(19,69)(20,98)(21,34)(22,72)(23,101)(24,37)(25,75)(26,104)(27,40)(29,94)(30,67)(32,97)(33,70)(35,100)(36,73)(38,103)(39,76)(41,106)(42,79)(44,82)(45,55)(47,85)(48,58)(50,88)(51,61)(53,91)(54,64)(56,84)(59,87)(62,90)(65,93)(68,96)(71,99)(74,102)(77,105)(80,108), (1,106)(3,108)(4,82)(6,84)(7,85)(9,87)(10,88)(12,90)(13,91)(15,93)(16,94)(18,96)(19,97)(21,99)(22,100)(24,102)(25,103)(27,105)(28,65)(29,66)(31,68)(32,69)(34,71)(35,72)(37,74)(38,75)(40,77)(41,78)(43,80)(44,81)(46,56)(47,57)(49,59)(50,60)(52,62)(53,63), (1,106)(2,107)(4,82)(5,83)(7,85)(8,86)(10,88)(11,89)(13,91)(14,92)(16,94)(17,95)(19,97)(20,98)(22,100)(23,101)(25,103)(26,104)(29,66)(30,67)(32,69)(33,70)(35,72)(36,73)(38,75)(39,76)(41,78)(42,79)(44,81)(45,55)(47,57)(48,58)(50,60)(51,61)(53,63)(54,64), (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)>;`

`G:=Group( (1,106)(2,42)(3,80)(4,82)(5,45)(6,56)(7,85)(8,48)(9,59)(10,88)(11,51)(12,62)(13,91)(14,54)(15,65)(16,94)(17,30)(18,68)(19,97)(20,33)(21,71)(22,100)(23,36)(24,74)(25,103)(26,39)(27,77)(28,93)(29,66)(31,96)(32,69)(34,99)(35,72)(37,102)(38,75)(40,105)(41,78)(43,108)(44,81)(46,84)(47,57)(49,87)(50,60)(52,90)(53,63)(55,83)(58,86)(61,89)(64,92)(67,95)(70,98)(73,101)(76,104)(79,107), (1,78)(2,107)(3,43)(4,81)(5,83)(6,46)(7,57)(8,86)(9,49)(10,60)(11,89)(12,52)(13,63)(14,92)(15,28)(16,66)(17,95)(18,31)(19,69)(20,98)(21,34)(22,72)(23,101)(24,37)(25,75)(26,104)(27,40)(29,94)(30,67)(32,97)(33,70)(35,100)(36,73)(38,103)(39,76)(41,106)(42,79)(44,82)(45,55)(47,85)(48,58)(50,88)(51,61)(53,91)(54,64)(56,84)(59,87)(62,90)(65,93)(68,96)(71,99)(74,102)(77,105)(80,108), (1,106)(3,108)(4,82)(6,84)(7,85)(9,87)(10,88)(12,90)(13,91)(15,93)(16,94)(18,96)(19,97)(21,99)(22,100)(24,102)(25,103)(27,105)(28,65)(29,66)(31,68)(32,69)(34,71)(35,72)(37,74)(38,75)(40,77)(41,78)(43,80)(44,81)(46,56)(47,57)(49,59)(50,60)(52,62)(53,63), (1,106)(2,107)(4,82)(5,83)(7,85)(8,86)(10,88)(11,89)(13,91)(14,92)(16,94)(17,95)(19,97)(20,98)(22,100)(23,101)(25,103)(26,104)(29,66)(30,67)(32,69)(33,70)(35,72)(36,73)(38,75)(39,76)(41,78)(42,79)(44,81)(45,55)(47,57)(48,58)(50,60)(51,61)(53,63)(54,64), (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) );`

`G=PermutationGroup([[(1,106),(2,42),(3,80),(4,82),(5,45),(6,56),(7,85),(8,48),(9,59),(10,88),(11,51),(12,62),(13,91),(14,54),(15,65),(16,94),(17,30),(18,68),(19,97),(20,33),(21,71),(22,100),(23,36),(24,74),(25,103),(26,39),(27,77),(28,93),(29,66),(31,96),(32,69),(34,99),(35,72),(37,102),(38,75),(40,105),(41,78),(43,108),(44,81),(46,84),(47,57),(49,87),(50,60),(52,90),(53,63),(55,83),(58,86),(61,89),(64,92),(67,95),(70,98),(73,101),(76,104),(79,107)], [(1,78),(2,107),(3,43),(4,81),(5,83),(6,46),(7,57),(8,86),(9,49),(10,60),(11,89),(12,52),(13,63),(14,92),(15,28),(16,66),(17,95),(18,31),(19,69),(20,98),(21,34),(22,72),(23,101),(24,37),(25,75),(26,104),(27,40),(29,94),(30,67),(32,97),(33,70),(35,100),(36,73),(38,103),(39,76),(41,106),(42,79),(44,82),(45,55),(47,85),(48,58),(50,88),(51,61),(53,91),(54,64),(56,84),(59,87),(62,90),(65,93),(68,96),(71,99),(74,102),(77,105),(80,108)], [(1,106),(3,108),(4,82),(6,84),(7,85),(9,87),(10,88),(12,90),(13,91),(15,93),(16,94),(18,96),(19,97),(21,99),(22,100),(24,102),(25,103),(27,105),(28,65),(29,66),(31,68),(32,69),(34,71),(35,72),(37,74),(38,75),(40,77),(41,78),(43,80),(44,81),(46,56),(47,57),(49,59),(50,60),(52,62),(53,63)], [(1,106),(2,107),(4,82),(5,83),(7,85),(8,86),(10,88),(11,89),(13,91),(14,92),(16,94),(17,95),(19,97),(20,98),(22,100),(23,101),(25,103),(26,104),(29,66),(30,67),(32,69),(33,70),(35,72),(36,73),(38,75),(39,76),(41,78),(42,79),(44,81),(45,55),(47,57),(48,58),(50,60),(51,61),(53,63),(54,64)], [(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)]])`

72 conjugacy classes

 class 1 2A ··· 2E 3A 3B 6A ··· 6J 9A ··· 9F 18A ··· 18AD 27A ··· 27R order 1 2 ··· 2 3 3 6 ··· 6 9 ··· 9 18 ··· 18 27 ··· 27 size 1 3 ··· 3 1 1 3 ··· 3 1 ··· 1 3 ··· 3 16 ··· 16

72 irreducible representations

 dim 1 1 1 1 3 3 3 type + + image C1 C3 C9 C27 A4 C3.A4 C9.A4 kernel C24⋊C27 C23×C18 C23×C6 C24 C2×C18 C2×C6 C22 # reps 1 2 6 18 5 10 30

Matrix representation of C24⋊C27 in GL6(𝔽109)

 108 0 0 0 0 0 0 1 0 0 0 0 0 0 108 0 0 0 0 0 0 1 0 0 0 0 0 95 108 0 0 0 0 33 0 108
,
 1 0 0 0 0 0 0 108 0 0 0 0 0 0 108 0 0 0 0 0 0 108 0 0 0 0 0 0 108 0 0 0 0 76 0 1
,
 108 0 0 0 0 0 0 108 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 108 0 0 0 0 0 0 1 0 0 0 0 0 0 108 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 75 0 0 0 0 0 0 0 0 102 108 0 0 0 0 87 7 1 0 0 0 21 38 0

`G:=sub<GL(6,GF(109))| [108,0,0,0,0,0,0,1,0,0,0,0,0,0,108,0,0,0,0,0,0,1,95,33,0,0,0,0,108,0,0,0,0,0,0,108],[1,0,0,0,0,0,0,108,0,0,0,0,0,0,108,0,0,0,0,0,0,108,0,76,0,0,0,0,108,0,0,0,0,0,0,1],[108,0,0,0,0,0,0,108,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[108,0,0,0,0,0,0,1,0,0,0,0,0,0,108,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,75,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,102,87,21,0,0,0,108,7,38,0,0,0,0,1,0] >;`

C24⋊C27 in GAP, Magma, Sage, TeX

`C_2^4\rtimes C_{27}`
`% in TeX`

`G:=Group("C2^4:C27");`
`// GroupNames label`

`G:=SmallGroup(432,226);`
`// by ID`

`G=gap.SmallGroup(432,226);`
`# by ID`

`G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,21,50,1515,2839,9077,15882]);`
`// Polycyclic`

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

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