(C2^2xC4).A4 - GroupNames

G = (C₂²×C₄).A₄ order 192 = 2⁶·3

4^th non-split extension by C₂²×C₄ of A₄ acting faithfully

Aliases: (C₂²×C₄).₄A₄, C₂³.₁₆(C₂×A₄), Q₈⋊A₄.₁C₂, (C₂²×Q₈).₁C₆, C₂².₃(C₄.A₄), C₂³.₇₈C₂³⋊C₃, C₂.₃(C₂⁴⋊C₆), SmallGroup(192,196)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₂²×Q₈ — (C₂²×C₄).A₄

Chief series

C₁ — C₂ — C₂³ — C₂²×Q₈ — Q₈⋊A₄ — (C₂²×C₄).A₄

Lower central

C₂²×Q₈ — (C₂²×C₄).A₄

Upper central

C₁ — C₂

Generators and relations for (C₂²×C₄).A₄
G = < a,b,c,d,e,f | a²=b²=c⁴=f³=1, d²=e²=c², faf^-1=ab=ba, dcd^-1=ac=ca, ad=da, ae=ea, ece^-1=bc=cb, bd=db, be=eb, fbf^-1=a, cf=fc, ede^-1=c²d, fdf^-1=c²de, fef^-1=d >

C₁

C₂

₃C₂

₁₆C₃

C₂²

₃C₂²

₄C₄

₆C₄

₁₆C₆

C₂³

₂Q₈

₃C₂×C₄

₆C₂×C₄

₆Q₈

₆C₂×C₄

₆Q₈

₆C₂×C₄

₄A₄

₁₆C₁₂

C₂²×C₄

₃C₂²×C₄

₆C₂×Q₈

₆C₄⋊C₄

₆C₂×Q₈

₆C₄⋊C₄

₄C₂×A₄

₈SL₂(𝔽₃)

C₂²×Q₈

₃C₂.C₄²

₃C₂×C₄⋊C₄

₄C₄×A₄

C₂³.₇₈C₂³

Q₈⋊A₄

(C₂²×C₄).A₄

Character table of (C₂²×C₄).A₄

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	4E	4F	6A	6B	12A	12B	12C	12D
size	1	1	3	3	16	16	4	4	12	12	12	12	16	16	16	16	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	-1	-1	-1	1	1	-1	1	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	ζ₃²	ζ₃	-1	-1	-1	1	1	-1	ζ₃	ζ₃²	ζ₆	ζ₆	ζ₆⁵	ζ₆⁵	linear of order 6
ρ₄	1	1	1	1	ζ₃	ζ₃²	-1	-1	-1	1	1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆⁵	ζ₆	ζ₆	linear of order 6
ρ₅	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃²	ζ₃	ζ₃	linear of order 3
ρ₆	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃	ζ₃²	ζ₃²	linear of order 3
ρ₇	2	-2	-2	2	-1	-1	2i	-2i	0	0	0	0	1	1	i	-i	i	-i	complex lifted from C₄.A₄
ρ₈	2	-2	-2	2	-1	-1	-2i	2i	0	0	0	0	1	1	-i	i	-i	i	complex lifted from C₄.A₄
ρ₉	2	-2	-2	2	ζ₆⁵	ζ₆	-2i	2i	0	0	0	0	ζ₃²	ζ₃	ζ₄³ζ₃	ζ₄ζ₃	ζ₄³ζ₃²	ζ₄ζ₃²	complex lifted from C₄.A₄
ρ₁₀	2	-2	-2	2	ζ₆	ζ₆⁵	2i	-2i	0	0	0	0	ζ₃	ζ₃²	ζ₄ζ₃²	ζ₄³ζ₃²	ζ₄ζ₃	ζ₄³ζ₃	complex lifted from C₄.A₄
ρ₁₁	2	-2	-2	2	ζ₆	ζ₆⁵	-2i	2i	0	0	0	0	ζ₃	ζ₃²	ζ₄³ζ₃²	ζ₄ζ₃²	ζ₄³ζ₃	ζ₄ζ₃	complex lifted from C₄.A₄
ρ₁₂	2	-2	-2	2	ζ₆⁵	ζ₆	2i	-2i	0	0	0	0	ζ₃²	ζ₃	ζ₄ζ₃	ζ₄³ζ₃	ζ₄ζ₃²	ζ₄³ζ₃²	complex lifted from C₄.A₄
ρ₁₃	3	3	3	3	0	0	-3	-3	1	-1	-1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₄	3	3	3	3	0	0	3	3	-1	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₅	6	6	-2	-2	0	0	0	0	0	-2	2	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₆
ρ₁₆	6	6	-2	-2	0	0	0	0	0	2	-2	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₆
ρ₁₇	6	-6	2	-2	0	0	0	0	2	0	0	-2	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₁₈	6	-6	2	-2	0	0	0	0	-2	0	0	2	0	0	0	0	0	0	symplectic faithful, Schur index 2

Permutation representations of (C₂²×C₄).A₄
►On 24 points - transitive group 24T304

Generators in S₂₄

(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)

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

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

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

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

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

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

G:=TransitiveGroup(24,304);

Matrix representation of (C₂²×C₄).A₄ ►in GL₆(𝔽₃)

1	2	1	2	1	2
0	2	0	0	0	0
2	0	1	2	2	0
0	2	2	2	2	2
1	2	0	1	2	2
0	0	0	0	0	2

2	0	1	2	1	2
0	2	1	2	1	2
2	2	1	0	1	0
1	1	1	2	2	0
0	0	2	1	1	1
1	1	1	0	2	2

0	0	1	1	2	1
0	1	0	0	0	1
0	2	1	1	0	1
0	1	1	2	0	1
1	1	2	0	0	1
0	1	0	0	0	2

1	2	1	1	1	1
1	2	0	1	1	0
1	1	1	1	0	2
1	1	2	2	1	1
2	1	0	1	2	1
1	0	2	2	0	1

2	1	2	2	1	2
0	0	1	2	1	1
2	2	0	2	0	0
1	2	1	2	1	0
0	2	0	2	1	2
2	0	1	1	0	1

0	1	2	1	0	0
1	0	0	1	2	0
2	1	0	1	0	0
1	1	0	2	0	0
1	2	0	1	0	0
1	0	0	2	0	1

G:=sub<GL(6,GF(3))| [1,0,2,0,1,0,2,2,0,2,2,0,1,0,1,2,0,0,2,0,2,2,1,0,1,0,2,2,2,0,2,0,0,2,2,2],[2,0,2,1,0,1,0,2,2,1,0,1,1,1,1,1,2,1,2,2,0,2,1,0,1,1,1,2,1,2,2,2,0,0,1,2],[0,0,0,0,1,0,0,1,2,1,1,1,1,0,1,1,2,0,1,0,1,2,0,0,2,0,0,0,0,0,1,1,1,1,1,2],[1,1,1,1,2,1,2,2,1,1,1,0,1,0,1,2,0,2,1,1,1,2,1,2,1,1,0,1,2,0,1,0,2,1,1,1],[2,0,2,1,0,2,1,0,2,2,2,0,2,1,0,1,0,1,2,2,2,2,2,1,1,1,0,1,1,0,2,1,0,0,2,1],[0,1,2,1,1,1,1,0,1,1,2,0,2,0,0,0,0,0,1,1,1,2,1,2,0,2,0,0,0,0,0,0,0,0,0,1] >;

(C₂²×C₄).A₄ in GAP, Magma, Sage, TeX

(C_2^2\times C_4).A_4

% in TeX

G:=Group("(C2^2xC4).A4");

// GroupNames label

G:=SmallGroup(192,196);

// by ID

G=gap.SmallGroup(192,196);

# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,632,135,352,1683,262,521,248,851,1524]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=f^3=1,d^2=e^2=c^2,f*a*f^-1=a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,a*e=e*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a,c*f=f*c,e*d*e^-1=c^2*d,f*d*f^-1=c^2*d*e,f*e*f^-1=d>;

// generators/relations

Export

Subgroup lattice of (C₂²×C₄).A₄ in TeX
Character table of (C₂²×C₄).A₄ in TeX

G = (C22×C4).A4 order 192 = 26·3

4th non-split extension by C22×C4 of A4 acting faithfully

G = (C₂²×C₄).A₄ order 192 = 2⁶·3

4^th non-split extension by C₂²×C₄ of A₄ acting faithfully