C2^2:A4 - GroupNames

G = C₂²⋊A₄ order 48 = 2⁴·3

The semidirect product of C₂² and A₄ acting via A₄/C₂²=C₃

metabelian, soluble, monomial, A-group

Aliases: C₂²⋊A₄, C₂⁴⋊₂C₃, SmallGroup(48,50)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂²⋊A₄

Chief series

C₁ — C₂² — C₂⁴ — C₂²⋊A₄

Lower central

C₂⁴ — C₂²⋊A₄

Upper central

C₁

Generators and relations for C₂²⋊A₄
G = < a,b,c,d,e | a²=b²=c²=d²=e³=1, eae^-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, ebe^-1=a, ece^-1=cd=dc, ede^-1=c >

C₁

₃C₂

₁₆C₃

C₂²

₃C₂²

₃C₂³

₄A₄

C₂⁴

C₂²⋊A₄

Character table of C₂²⋊A₄

class	1	2A	2B	2C	2D	2E	3A	3B
size	1	3	3	3	3	3	16	16
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	ζ₃	ζ₃²	linear of order 3
ρ₃	1	1	1	1	1	1	ζ₃²	ζ₃	linear of order 3
ρ₄	3	3	-1	-1	-1	-1	0	0	orthogonal lifted from A₄
ρ₅	3	-1	-1	-1	3	-1	0	0	orthogonal lifted from A₄
ρ₆	3	-1	3	-1	-1	-1	0	0	orthogonal lifted from A₄
ρ₇	3	-1	-1	3	-1	-1	0	0	orthogonal lifted from A₄
ρ₈	3	-1	-1	-1	-1	3	0	0	orthogonal lifted from A₄

Permutation representations of C₂²⋊A₄
►On 12 points - transitive group 12T32

Generators in S₁₂

(1 10)(3 12)(4 8)(6 7)

(1 10)(2 11)(4 8)(5 9)

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

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

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

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

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

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

G:=TransitiveGroup(12,32);

►On 16 points - transitive group 16T64

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,64);

►On 24 points - transitive group 24T59

Generators in S₂₄

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,59);

C₂²⋊A₄ is a maximal subgroup of
C₂⁴⋊C₆ C₂²⋊S₄ A₄² C₄²⋊₂A₄ C₄²⋊A₄ C₂⁶⋊C₃ F₁₆ C₇⋊(C₂²⋊A₄)
C₂²⋊A₄ is a maximal quotient of
Q₈⋊A₄ C₂³⋊A₄ C₂⁴⋊C₉ C₄²⋊₂A₄ C₄²⋊A₄ C₄².A₄ C₂⁶⋊C₃ C₇⋊(C₂²⋊A₄)

Polynomial with Galois group C₂²⋊A₄ over ℚ

action	f(x)	Disc(f)
12T32	x¹²-47x¹⁰+646x⁸-2800x⁶+2584x⁴-752x²+64	2⁷⁰·3¹²·19⁴·307⁸

Matrix representation of C₂²⋊A₄ ►in GL₆(ℤ)

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	-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

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

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

C₂²⋊A₄ in GAP, Magma, Sage, TeX

C_2^2\rtimes A_4

% in TeX

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

// GroupNames label

G:=SmallGroup(48,50);

// by ID

G=gap.SmallGroup(48,50);

# by ID

G:=PCGroup([5,-3,-2,2,-2,2,61,137,483,904]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂²⋊A₄ in TeX
Character table of C₂²⋊A₄ in TeX

G = C22⋊A4 order 48 = 24·3

The semidirect product of C22 and A4 acting via A4/C22=C3

G = C₂²⋊A₄ order 48 = 2⁴·3

The semidirect product of C₂² and A₄ acting via A₄/C₂²=C₃