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G = C22⋊A4order 48 = 24·3

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

metabelian, soluble, monomial, A-group

Aliases: C22⋊A4, C242C3, SmallGroup(48,50)

Series: Derived Chief Lower central Upper central

C1C24 — C22⋊A4
C1C22C24 — C22⋊A4
C24 — C22⋊A4
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 12A2B2C2D2E3A3B
 size 1333331616
ρ111111111    trivial
ρ2111111ζ3ζ32    linear of order 3
ρ3111111ζ32ζ3    linear of order 3
ρ433-1-1-1-100    orthogonal lifted from A4
ρ53-1-1-13-100    orthogonal lifted from A4
ρ63-13-1-1-100    orthogonal lifted from A4
ρ73-1-13-1-100    orthogonal lifted from A4
ρ83-1-1-1-1300    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 13)(2 8)(3 15)(4 6)(5 9)(7 14)(10 16)(11 12)
(1 11)(2 7)(3 9)(4 16)(5 15)(6 10)(8 14)(12 13)
(1 7)(2 11)(3 16)(4 9)(5 6)(8 12)(10 15)(13 14)
(1 5)(2 10)(3 12)(4 14)(6 7)(8 16)(9 13)(11 15)
(2 3 4)(5 6 7)(8 9 10)(11 12 13)(14 15 16)

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

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

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

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)

100000
010000
001000
000-100
0000-10
000001
,
100000
010000
001000
000-100
000010
00000-1
,
100000
0-10000
00-1000
000100
000010
000001
,
-100000
0-10000
001000
000100
000010
000001
,
010000
001000
100000
000010
000001
000100

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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