Copied to
clipboard

## G = C23.A4order 96 = 25·3

### 2nd non-split extension by C23 of A4 acting faithfully

Aliases: C422C6, C23.2A4, C41D4⋊C3, C42⋊C33C2, C22.4(C2×A4), SmallGroup(96,72)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C23.A4
 Chief series C1 — C22 — C42 — C42⋊C3 — C23.A4
 Lower central C42 — C23.A4
 Upper central C1

Generators and relations for C23.A4
G = < a,b,c,d,e,f | a2=b2=c2=f3=1, d2=cb=fbf-1=bc, e2=fcf-1=b, eae-1=ab=ba, ac=ca, dad-1=abc, af=fa, bd=db, be=eb, fef-1=cd=dc, ce=ec, de=ed, fdf-1=bde >

3C2
4C2
12C2
16C3
3C4
3C4
6C22
6C22
6C22
6C22
16C6
3C23
6D4
6D4
6D4
6D4
4A4

Character table of C23.A4

 class 1 2A 2B 2C 3A 3B 4A 4B 6A 6B size 1 3 4 12 16 16 6 6 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 linear of order 3 ρ5 1 1 -1 -1 ζ3 ζ32 1 1 ζ6 ζ65 linear of order 6 ρ6 1 1 -1 -1 ζ32 ζ3 1 1 ζ65 ζ6 linear of order 6 ρ7 3 3 3 -1 0 0 -1 -1 0 0 orthogonal lifted from A4 ρ8 3 3 -3 1 0 0 -1 -1 0 0 orthogonal lifted from C2×A4 ρ9 6 -2 0 0 0 0 2 -2 0 0 orthogonal faithful ρ10 6 -2 0 0 0 0 -2 2 0 0 orthogonal faithful

Permutation representations of C23.A4
On 12 points - transitive group 12T60
Generators in S12
```(1 3)(2 4)(5 8)(6 7)(9 10)(11 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 6 7 8)(9 11)(10 12)
(1 9 5)(2 11 7)(3 10 8)(4 12 6)```

`G:=sub<Sym(12)| (1,3)(2,4)(5,8)(6,7)(9,10)(11,12), (1,2)(3,4)(5,7)(6,8), (1,2)(3,4)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,4,2,3)(5,6,7,8)(9,11)(10,12), (1,9,5)(2,11,7)(3,10,8)(4,12,6)>;`

`G:=Group( (1,3)(2,4)(5,8)(6,7)(9,10)(11,12), (1,2)(3,4)(5,7)(6,8), (1,2)(3,4)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,4,2,3)(5,6,7,8)(9,11)(10,12), (1,9,5)(2,11,7)(3,10,8)(4,12,6) );`

`G=PermutationGroup([(1,3),(2,4),(5,8),(6,7),(9,10),(11,12)], [(1,2),(3,4),(5,7),(6,8)], [(1,2),(3,4),(9,11),(10,12)], [(5,6,7,8),(9,10,11,12)], [(1,4,2,3),(5,6,7,8),(9,11),(10,12)], [(1,9,5),(2,11,7),(3,10,8),(4,12,6)])`

`G:=TransitiveGroup(12,60);`

On 12 points - transitive group 12T61
Generators in S12
```(3 4)(6 8)(10 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(9 11)(10 12)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 8 7 6)
(1 11 5)(2 9 7)(3 10 8)(4 12 6)```

`G:=sub<Sym(12)| (3,4)(6,8)(10,12), (1,2)(3,4)(5,7)(6,8), (1,2)(3,4)(9,11)(10,12), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,4,2,3)(5,8,7,6), (1,11,5)(2,9,7)(3,10,8)(4,12,6)>;`

`G:=Group( (3,4)(6,8)(10,12), (1,2)(3,4)(5,7)(6,8), (1,2)(3,4)(9,11)(10,12), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,4,2,3)(5,8,7,6), (1,11,5)(2,9,7)(3,10,8)(4,12,6) );`

`G=PermutationGroup([(3,4),(6,8),(10,12)], [(1,2),(3,4),(5,7),(6,8)], [(1,2),(3,4),(9,11),(10,12)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,4,2,3),(5,8,7,6)], [(1,11,5),(2,9,7),(3,10,8),(4,12,6)])`

`G:=TransitiveGroup(12,61);`

On 16 points - transitive group 16T185
Generators in S16
```(2 4)(5 12)(6 11)(7 10)(8 9)(14 16)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 13 5)(2 9 14 6)(3 10 15 7)(4 11 16 8)
(2 10 6)(3 13 15)(4 7 11)(5 8 14)(9 16 12)```

`G:=sub<Sym(16)| (2,4)(5,12)(6,11)(7,10)(8,9)(14,16), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12)>;`

`G:=Group( (2,4)(5,12)(6,11)(7,10)(8,9)(14,16), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12) );`

`G=PermutationGroup([(2,4),(5,12),(6,11),(7,10),(8,9),(14,16)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)], [(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,12,13,5),(2,9,14,6),(3,10,15,7),(4,11,16,8)], [(2,10,6),(3,13,15),(4,7,11),(5,8,14),(9,16,12)])`

`G:=TransitiveGroup(16,185);`

On 24 points - transitive group 24T187
Generators in S24
```(1 6)(2 5)(3 7)(4 8)(9 21)(10 24)(11 23)(12 22)(13 18)(14 17)(15 20)(16 19)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(21 24 23 22)
(1 14 24)(2 16 22)(3 13 23)(4 15 21)(5 19 12)(6 17 10)(7 18 11)(8 20 9)```

`G:=sub<Sym(24)| (1,6)(2,5)(3,7)(4,8)(9,21)(10,24)(11,23)(12,22)(13,18)(14,17)(15,20)(16,19), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,8,6,7)(9,12,11,10)(21,24,23,22), (1,14,24)(2,16,22)(3,13,23)(4,15,21)(5,19,12)(6,17,10)(7,18,11)(8,20,9)>;`

`G:=Group( (1,6)(2,5)(3,7)(4,8)(9,21)(10,24)(11,23)(12,22)(13,18)(14,17)(15,20)(16,19), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,8,6,7)(9,12,11,10)(21,24,23,22), (1,14,24)(2,16,22)(3,13,23)(4,15,21)(5,19,12)(6,17,10)(7,18,11)(8,20,9) );`

`G=PermutationGroup([(1,6),(2,5),(3,7),(4,8),(9,21),(10,24),(11,23),(12,22),(13,18),(14,17),(15,20),(16,19)], [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(13,15),(14,16),(17,19),(18,20)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,3),(5,8,6,7),(9,12,11,10),(21,24,23,22)], [(1,14,24),(2,16,22),(3,13,23),(4,15,21),(5,19,12),(6,17,10),(7,18,11),(8,20,9)])`

`G:=TransitiveGroup(24,187);`

On 24 points - transitive group 24T188
Generators in S24
```(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 17)(14 20)(15 19)(16 18)(21 22)(23 24)
(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(21 23)(22 24)
(1 6)(2 5)(3 8)(4 7)(13 15)(14 16)(17 19)(18 20)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3 6 8)(2 4 5 7)(9 23 11 21)(10 24 12 22)(13 20)(14 17)(15 18)(16 19)
(1 16 11)(2 17 24)(3 20 12)(4 15 21)(5 19 22)(6 14 9)(7 13 23)(8 18 10)```

`G:=sub<Sym(24)| (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,17)(14,20)(15,19)(16,18)(21,22)(23,24), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(21,23)(22,24), (1,6)(2,5)(3,8)(4,7)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,6,8)(2,4,5,7)(9,23,11,21)(10,24,12,22)(13,20)(14,17)(15,18)(16,19), (1,16,11)(2,17,24)(3,20,12)(4,15,21)(5,19,22)(6,14,9)(7,13,23)(8,18,10)>;`

`G:=Group( (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,17)(14,20)(15,19)(16,18)(21,22)(23,24), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(21,23)(22,24), (1,6)(2,5)(3,8)(4,7)(13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,6,8)(2,4,5,7)(9,23,11,21)(10,24,12,22)(13,20)(14,17)(15,18)(16,19), (1,16,11)(2,17,24)(3,20,12)(4,15,21)(5,19,22)(6,14,9)(7,13,23)(8,18,10) );`

`G=PermutationGroup([(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,17),(14,20),(15,19),(16,18),(21,22),(23,24)], [(1,6),(2,5),(3,8),(4,7),(9,11),(10,12),(21,23),(22,24)], [(1,6),(2,5),(3,8),(4,7),(13,15),(14,16),(17,19),(18,20)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3,6,8),(2,4,5,7),(9,23,11,21),(10,24,12,22),(13,20),(14,17),(15,18),(16,19)], [(1,16,11),(2,17,24),(3,20,12),(4,15,21),(5,19,22),(6,14,9),(7,13,23),(8,18,10)])`

`G:=TransitiveGroup(24,188);`

On 24 points - transitive group 24T189
Generators in S24
```(5 8)(6 7)(10 12)(14 16)(17 19)(21 23)
(1 4)(2 3)(5 8)(6 7)(9 11)(10 12)(17 19)(18 20)
(1 4)(2 3)(5 8)(6 7)(13 15)(14 16)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 6 4 7)(2 5 3 8)(9 17 11 19)(10 18 12 20)(13 22)(14 23)(15 24)(16 21)
(1 24 20)(2 13 9)(3 15 11)(4 22 18)(5 23 10)(6 16 17)(7 14 19)(8 21 12)```

`G:=sub<Sym(24)| (5,8)(6,7)(10,12)(14,16)(17,19)(21,23), (1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(17,19)(18,20), (1,4)(2,3)(5,8)(6,7)(13,15)(14,16)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,4,7)(2,5,3,8)(9,17,11,19)(10,18,12,20)(13,22)(14,23)(15,24)(16,21), (1,24,20)(2,13,9)(3,15,11)(4,22,18)(5,23,10)(6,16,17)(7,14,19)(8,21,12)>;`

`G:=Group( (5,8)(6,7)(10,12)(14,16)(17,19)(21,23), (1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(17,19)(18,20), (1,4)(2,3)(5,8)(6,7)(13,15)(14,16)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,4,7)(2,5,3,8)(9,17,11,19)(10,18,12,20)(13,22)(14,23)(15,24)(16,21), (1,24,20)(2,13,9)(3,15,11)(4,22,18)(5,23,10)(6,16,17)(7,14,19)(8,21,12) );`

`G=PermutationGroup([(5,8),(6,7),(10,12),(14,16),(17,19),(21,23)], [(1,4),(2,3),(5,8),(6,7),(9,11),(10,12),(17,19),(18,20)], [(1,4),(2,3),(5,8),(6,7),(13,15),(14,16),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6,4,7),(2,5,3,8),(9,17,11,19),(10,18,12,20),(13,22),(14,23),(15,24),(16,21)], [(1,24,20),(2,13,9),(3,15,11),(4,22,18),(5,23,10),(6,16,17),(7,14,19),(8,21,12)])`

`G:=TransitiveGroup(24,189);`

On 24 points - transitive group 24T190
Generators in S24
```(1 2)(3 4)(5 8)(6 7)(9 15)(10 14)(11 13)(12 16)(17 21)(18 24)(19 23)(20 22)
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 4)(2 3)(5 7)(6 8)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 7 4 5)(2 8 3 6)(9 16 11 14)(10 13 12 15)(17 23)(18 24)(19 21)(20 22)
(1 23 9)(2 19 15)(3 17 13)(4 21 11)(5 20 12)(6 24 14)(7 18 10)(8 22 16)```

`G:=sub<Sym(24)| (1,2)(3,4)(5,8)(6,7)(9,15)(10,14)(11,13)(12,16)(17,21)(18,24)(19,23)(20,22), (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,4)(2,3)(5,7)(6,8)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,7,4,5)(2,8,3,6)(9,16,11,14)(10,13,12,15)(17,23)(18,24)(19,21)(20,22), (1,23,9)(2,19,15)(3,17,13)(4,21,11)(5,20,12)(6,24,14)(7,18,10)(8,22,16)>;`

`G:=Group( (1,2)(3,4)(5,8)(6,7)(9,15)(10,14)(11,13)(12,16)(17,21)(18,24)(19,23)(20,22), (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,4)(2,3)(5,7)(6,8)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,7,4,5)(2,8,3,6)(9,16,11,14)(10,13,12,15)(17,23)(18,24)(19,21)(20,22), (1,23,9)(2,19,15)(3,17,13)(4,21,11)(5,20,12)(6,24,14)(7,18,10)(8,22,16) );`

`G=PermutationGroup([(1,2),(3,4),(5,8),(6,7),(9,15),(10,14),(11,13),(12,16),(17,21),(18,24),(19,23),(20,22)], [(1,4),(2,3),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,4),(2,3),(5,7),(6,8),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,7,4,5),(2,8,3,6),(9,16,11,14),(10,13,12,15),(17,23),(18,24),(19,21),(20,22)], [(1,23,9),(2,19,15),(3,17,13),(4,21,11),(5,20,12),(6,24,14),(7,18,10),(8,22,16)])`

`G:=TransitiveGroup(24,190);`

C23.A4 is a maximal subgroup of   C42⋊Dic3  C42⋊D6  C24.6A4  (C4×C12)⋊C6  C204D4⋊C3
C23.A4 is a maximal quotient of   C422C12  C232D4⋊C3  C24.3A4  C422C18  (C4×C12)⋊C6  C204D4⋊C3

Polynomial with Galois group C23.A4 over ℚ
actionf(x)Disc(f)
12T60x12-14x8-7x4+4254·318
12T61x12-11x8+30x4-16-244·318

Matrix representation of C23.A4 in GL6(ℤ)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 -1 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 0 0 0 0 0 0 1
,
 -1 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 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0
,
 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0

`G:=sub<GL(6,Integers())| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C23.A4 in GAP, Magma, Sage, TeX

`C_2^3.A_4`
`% in TeX`

`G:=Group("C2^3.A4");`
`// GroupNames label`

`G:=SmallGroup(96,72);`
`// by ID`

`G=gap.SmallGroup(96,72);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,2,-2,2,1406,116,230,867,801,69,730,1307]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=f^3=1,d^2=c*b=f*b*f^-1=b*c,e^2=f*c*f^-1=b,e*a*e^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*f=f*a,b*d=d*b,b*e=e*b,f*e*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,f*d*f^-1=b*d*e>;`
`// generators/relations`

Export

׿
×
𝔽