C2^3.3A4 - GroupNames

G = C₂³.₃A₄ order 96 = 2⁵·3

1^st non-split extension by C₂³ of A₄ acting via A₄/C₂²=C₃

non-abelian, soluble

Aliases: C₂³.₃A₄, C₂².SL₂(𝔽₃), C₂.(C₄²⋊C₃), C₂.C₄²⋊C₃, SmallGroup(96,3)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₂.C₄² — C₂³.₃A₄

Chief series

C₁ — C₂ — C₂³ — C₂.C₄² — C₂³.₃A₄

Lower central

C₂.C₄² — C₂³.₃A₄

Upper central

C₁ — C₂

Generators and relations for C₂³.₃A₄
G = < a,b,c,d,e,f | a²=b²=c²=f³=1, d²=faf^-1=abc, e²=fbf^-1=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede^-1=cd=dc, ce=ec, cf=fc, fdf^-1=ade, fef^-1=bcd >

C₁

C₂

₃C₂

₁₆C₃

C₂²

₃C₂²

₆C₄

₁₆C₆

C₂³

₃C₂×C₄

₆C₂×C₄

₄A₄

₃C₂²×C₄

₄C₂×A₄

C₂.C₄²

C₂³.₃A₄

Character table of C₂³.₃A₄

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	6A	6B
size	1	1	3	3	16	16	6	6	6	6	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	ζ₃²	ζ₃	linear of order 3
ρ₃	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	ζ₃	ζ₃²	linear of order 3
ρ₄	2	-2	-2	2	-1	-1	0	0	0	0	1	1	symplectic lifted from SL₂(𝔽₃), Schur index 2
ρ₅	2	-2	-2	2	ζ₆	ζ₆⁵	0	0	0	0	ζ₃	ζ₃²	complex lifted from SL₂(𝔽₃)
ρ₆	2	-2	-2	2	ζ₆⁵	ζ₆	0	0	0	0	ζ₃²	ζ₃	complex lifted from SL₂(𝔽₃)
ρ₇	3	3	3	3	0	0	-1	-1	-1	-1	0	0	orthogonal lifted from A₄
ρ₈	3	3	-1	-1	0	0	1	-1-2i	1	-1+2i	0	0	complex lifted from C₄²⋊C₃
ρ₉	3	3	-1	-1	0	0	1	-1+2i	1	-1-2i	0	0	complex lifted from C₄²⋊C₃
ρ₁₀	3	3	-1	-1	0	0	-1+2i	1	-1-2i	1	0	0	complex lifted from C₄²⋊C₃
ρ₁₁	3	3	-1	-1	0	0	-1-2i	1	-1+2i	1	0	0	complex lifted from C₄²⋊C₃
ρ₁₂	6	-6	2	-2	0	0	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₂³.₃A₄
►On 12 points - transitive group 12T57

Generators in S₁₂

(1 2)(5 6)

(3 7)(4 8)

(1 2)(3 7)(4 8)(5 6)(9 11)(10 12)

(3 4)(5 6)(7 8)(9 10 11 12)

(1 5 2 6)(3 7)(9 12)(10 11)

(1 7 12)(2 3 10)(4 11 5)(6 8 9)

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

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

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

G:=TransitiveGroup(12,57);

►On 24 points - transitive group 24T179

Generators in S₂₄

(1 3)(2 4)(5 7)(6 8)

(9 12)(10 11)(13 15)(14 16)

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

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

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

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

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

G:=TransitiveGroup(24,179);

►On 24 points - transitive group 24T180

Generators in S₂₄

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

(1 8)(2 7)(3 4)(5 6)(9 23)(10 24)(11 21)(12 22)

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

(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 3)(4 8 5 7)(9 21)(11 23)(13 16 20 19)(14 18 17 15)

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

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

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

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

G:=TransitiveGroup(24,180);

C₂³.₃A₄ is a maximal subgroup of C₂³.₇S₄ C₂³.₈S₄ C₄²⋊₄C₄⋊C₃ C₂³⋊₂D₄⋊C₃ C₂⁴.₂A₄ C₂⁴.₃A₄ C₂³.₁₉(C₂×A₄)
C₂³.₃A₄ is a maximal quotient of C₂³.SL₂(𝔽₃) C₂.(C₄²⋊C₉)

Polynomial with Galois group C₂³.₃A₄ over ℚ

action	f(x)	Disc(f)
12T57	x¹²-69x¹⁰-2091x⁸-7571x⁶+134691x⁴+960267x²+1545049	2¹²·3²⁸·7¹²·11¹⁰·13⁴·109¹²·113²·127⁸

Matrix representation of C₂³.₃A₄ ►in GL₅(𝔽₁₃)

12	0	0	0	0
0	12	0	0	0
0	0	12	0	0
0	0	0	1	0
0	0	0	0	12

12	0	0	0	0
0	12	0	0	0
0	0	1	0	0
0	0	0	12	0
0	0	0	0	12

12	0	0	0	0
0	12	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	1	0	0	0
12	0	0	0	0
0	0	5	0	0
0	0	0	8	0
0	0	0	0	1

4	3	0	0	0
3	9	0	0	0
0	0	5	0	0
0	0	0	12	0
0	0	0	0	5

1	0	0	0	0
3	9	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,12,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,1],[4,3,0,0,0,3,9,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,5],[1,3,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C₂³.₃A₄ in GAP, Magma, Sage, TeX

C_2^3._3A_4

% in TeX

G:=Group("C2^3.3A4");

// GroupNames label

G:=SmallGroup(96,3);

// by ID

G=gap.SmallGroup(96,3);

# by ID

G:=PCGroup([6,-3,-2,2,-2,2,-2,73,151,596,332,50,1731,2524]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂³.₃A₄ in TeX
Character table of C₂³.₃A₄ in TeX

G = C23.3A4 order 96 = 25·3

1st non-split extension by C23 of A4 acting via A4/C22=C3

G = C₂³.₃A₄ order 96 = 2⁵·3

1^st non-split extension by C₂³ of A₄ acting via A₄/C₂²=C₃