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## G = C22⋊A4order 48 = 24·3

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

Aliases: C22⋊A4, C242C3, 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 >

3C2
3C2
3C2
3C2
3C2
16C3
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C23
3C23
3C23
3C23
3C23
4A4
4A4
4A4
4A4
4A4

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  C422A4  C42⋊A4  C26⋊C3  F16  C7⋊(C22⋊A4)
C22⋊A4 is a maximal quotient of
Q8⋊A4  C23⋊A4  C24⋊C9  C422A4  C42⋊A4  C42.A4  C26⋊C3  C7⋊(C22⋊A4)

Polynomial with Galois group C22⋊A4 over ℚ
actionf(x)Disc(f)
12T32x12-47x10+646x8-2800x6+2584x4-752x2+64270·312·194·3078

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

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