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## G = C43order 64 = 26

### Abelian group of type [4,4,4]

Aliases: C43, SmallGroup(64,55)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C43
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43
 Lower central C1 — C43
 Upper central C1 — C43
 Jennings C1 — C23 — C43

Generators and relations for C43
G = < a,b,c | a4=b4=c4=1, ab=ba, ac=ca, bc=cb >

Subgroups: 129, all normal (3 characteristic)
C1, C2 [×7], C4 [×28], C22 [×7], C2×C4 [×42], C23, C42 [×28], C22×C4 [×7], C2×C42 [×7], C43
Quotients: C1, C2 [×7], C4 [×28], C22 [×7], C2×C4 [×42], C23, C42 [×28], C22×C4 [×7], C2×C42 [×7], C43

Smallest permutation representation of C43
Regular action on 64 points
Generators in S64
```(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)
(1 49 41 61)(2 50 42 62)(3 51 43 63)(4 52 44 64)(5 60 20 12)(6 57 17 9)(7 58 18 10)(8 59 19 11)(13 34 25 21)(14 35 26 22)(15 36 27 23)(16 33 28 24)(29 53 45 38)(30 54 46 39)(31 55 47 40)(32 56 48 37)
(1 47 11 25)(2 48 12 26)(3 45 9 27)(4 46 10 28)(5 22 50 37)(6 23 51 38)(7 24 52 39)(8 21 49 40)(13 41 31 59)(14 42 32 60)(15 43 29 57)(16 44 30 58)(17 36 63 53)(18 33 64 54)(19 34 61 55)(20 35 62 56)```

`G:=sub<Sym(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), (1,49,41,61)(2,50,42,62)(3,51,43,63)(4,52,44,64)(5,60,20,12)(6,57,17,9)(7,58,18,10)(8,59,19,11)(13,34,25,21)(14,35,26,22)(15,36,27,23)(16,33,28,24)(29,53,45,38)(30,54,46,39)(31,55,47,40)(32,56,48,37), (1,47,11,25)(2,48,12,26)(3,45,9,27)(4,46,10,28)(5,22,50,37)(6,23,51,38)(7,24,52,39)(8,21,49,40)(13,41,31,59)(14,42,32,60)(15,43,29,57)(16,44,30,58)(17,36,63,53)(18,33,64,54)(19,34,61,55)(20,35,62,56)>;`

`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), (1,49,41,61)(2,50,42,62)(3,51,43,63)(4,52,44,64)(5,60,20,12)(6,57,17,9)(7,58,18,10)(8,59,19,11)(13,34,25,21)(14,35,26,22)(15,36,27,23)(16,33,28,24)(29,53,45,38)(30,54,46,39)(31,55,47,40)(32,56,48,37), (1,47,11,25)(2,48,12,26)(3,45,9,27)(4,46,10,28)(5,22,50,37)(6,23,51,38)(7,24,52,39)(8,21,49,40)(13,41,31,59)(14,42,32,60)(15,43,29,57)(16,44,30,58)(17,36,63,53)(18,33,64,54)(19,34,61,55)(20,35,62,56) );`

`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)], [(1,49,41,61),(2,50,42,62),(3,51,43,63),(4,52,44,64),(5,60,20,12),(6,57,17,9),(7,58,18,10),(8,59,19,11),(13,34,25,21),(14,35,26,22),(15,36,27,23),(16,33,28,24),(29,53,45,38),(30,54,46,39),(31,55,47,40),(32,56,48,37)], [(1,47,11,25),(2,48,12,26),(3,45,9,27),(4,46,10,28),(5,22,50,37),(6,23,51,38),(7,24,52,39),(8,21,49,40),(13,41,31,59),(14,42,32,60),(15,43,29,57),(16,44,30,58),(17,36,63,53),(18,33,64,54),(19,34,61,55),(20,35,62,56)])`

C43 is a maximal subgroup of
C426C8  C424C8  C43.C2  C43.7C2  C428C8  C425C8  C429C8  C43⋊C2  C4216Q8  C439C2  C4214Q8  C432C2  C4312C2  C43.15C2  C4313C2  C4314C2  C4218Q8  C4215Q8  C43.18C2  C434C2  C435C2  C4315C2  C4219Q8  C43⋊C7
C43 is a maximal quotient of
C2.C43

64 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4BD order 1 2 ··· 2 4 ··· 4 size 1 1 ··· 1 1 ··· 1

64 irreducible representations

 dim 1 1 1 type + + image C1 C2 C4 kernel C43 C2×C42 C42 # reps 1 7 56

Matrix representation of C43 in GL3(𝔽5) generated by

 3 0 0 0 4 0 0 0 2
,
 3 0 0 0 2 0 0 0 1
,
 4 0 0 0 1 0 0 0 3
`G:=sub<GL(3,GF(5))| [3,0,0,0,4,0,0,0,2],[3,0,0,0,2,0,0,0,1],[4,0,0,0,1,0,0,0,3] >;`

C43 in GAP, Magma, Sage, TeX

`C_4^3`
`% in TeX`

`G:=Group("C4^3");`
`// GroupNames label`

`G:=SmallGroup(64,55);`
`// by ID`

`G=gap.SmallGroup(64,55);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,2,48,103,158]);`
`// Polycyclic`

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

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