C2^3.A4 - GroupNames

G = C₂³.A₄ order 96 = 2⁵·3

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

metabelian, soluble, monomial

Aliases: C₄²⋊₂C₆, C₂³.₂A₄, C₄⋊₁D₄⋊C₃, C₄²⋊C₃⋊₃C₂, 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 >

C₁

₃C₂

₄C₂

₁₂C₂

₁₆C₃

C₂²

₃C₄

₆C₂²

₁₆C₆

C₂³

₃C₂×C₄

₃C₂³

₆D₄

₄A₄

C₄²

₃C₂×D₄

₄C₂×A₄

C₄⋊₁D₄

C₄²⋊C₃

C₂³.A₄

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 7)(2 8)(3 5)(4 6)(9 21)(10 24)(11 23)(12 22)(13 20)(14 19)(15 18)(16 17)

(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 11)(10 12)(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 2 5)(3 8 4 7)(13 16 15 14)(17 20 19 18)

(1 23 20)(2 21 18)(3 12 14)(4 10 16)(5 22 19)(6 24 17)(7 11 13)(8 9 15)

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

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

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

G:=TransitiveGroup(24,187);

►On 24 points - transitive group 24T188

Generators in S₂₄

(1 7)(2 8)(3 5)(4 6)(9 12)(10 11)(13 18)(14 17)(15 20)(16 19)(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 4 6 7)(2 3 5 8)(9 21 11 23)(10 22 12 24)(13 17)(14 18)(15 19)(16 20)

(1 14 22)(2 20 11)(3 13 12)(4 19 23)(5 18 9)(6 16 24)(7 17 21)(8 15 10)

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

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

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

G:=TransitiveGroup(24,188);

►On 24 points - transitive group 24T189

Generators in S₂₄

(1 6)(2 5)(10 12)(14 16)(18 20)(21 23)

(1 6)(2 5)(3 8)(4 7)(17 19)(18 20)(21 23)(22 24)

(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)

(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 6 4)(2 8 5 3)(9 15)(10 16)(11 13)(12 14)(17 23 19 21)(18 24 20 22)

(1 12 20)(2 16 21)(3 9 24)(4 13 19)(5 14 23)(6 10 18)(7 15 17)(8 11 22)

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

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

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

G:=TransitiveGroup(24,189);

►On 24 points - transitive group 24T190

Generators in S₂₄

(1 2)(3 4)(5 8)(6 7)(9 24)(10 23)(11 22)(12 21)(13 18)(14 17)(15 20)(16 19)

(1 4)(2 3)(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)

(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(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 24)(10 21)(11 22)(12 23)(13 19 15 17)(14 20 16 18)

(1 23 18)(2 10 13)(3 12 15)(4 21 20)(5 11 17)(6 24 16)(7 9 19)(8 22 14)

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

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

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

G:=TransitiveGroup(24,190);

C₂³.A₄ is a maximal subgroup of C₄²⋊Dic₃ C₄²⋊D₆ C₂⁴.₆A₄ (C₄×C₁₂)⋊C₆ C₂₀⋊₄D₄⋊C₃
C₂³.A₄ is a maximal quotient of C₄²⋊₂C₁₂ C₂³⋊₂D₄⋊C₃ C₂⁴.₃A₄ C₄²⋊₂C₁₈ (C₄×C₁₂)⋊C₆ C₂₀⋊₄D₄⋊C₃

Polynomial with Galois group C₂³.A₄ over ℚ

action	f(x)	Disc(f)
12T60	x¹²-14x⁸-7x⁴+4	2⁵⁴·31⁸
12T61	x¹²-11x⁸+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

Subgroup lattice of C₂³.A₄ in TeX
Character table of C₂³.A₄ in TeX

G = C23.A4 order 96 = 25·3

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

G = C₂³.A₄ order 96 = 2⁵·3

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