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## G = C2×C22⋊A4order 96 = 25·3

### Direct product of C2 and C22⋊A4

Aliases: C2×C22⋊A4, C252C3, C233A4, C244C6, C22⋊(C2×A4), SmallGroup(96,229)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C2×C22⋊A4
 Chief series C1 — C22 — C24 — C22⋊A4 — C2×C22⋊A4
 Lower central C24 — C2×C22⋊A4
 Upper central C1 — C2

Generators and relations for C2×C22⋊A4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >

Subgroups: 448 in 148 conjugacy classes, 16 normal (6 characteristic)
C1, C2, C2, C3, C22, C22, C6, C23, C23, A4, C24, C24, C2×A4, C25, C22⋊A4, C2×C22⋊A4
Quotients: C1, C2, C3, C6, A4, C2×A4, C22⋊A4, C2×C22⋊A4

Character table of C2×C22⋊A4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 6A 6B size 1 1 3 3 3 3 3 3 3 3 3 3 16 16 16 16 ρ1 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 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 ζ32 ζ3 ζ6 ζ65 linear of order 6 ρ5 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 ζ3 ζ32 ζ65 ζ6 linear of order 6 ρ6 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ7 3 3 -1 3 -1 -1 -1 -1 -1 3 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ8 3 -3 -1 -1 3 -1 -1 1 1 1 -3 1 0 0 0 0 orthogonal lifted from C2×A4 ρ9 3 -3 -1 3 -1 -1 -1 1 1 -3 1 1 0 0 0 0 orthogonal lifted from C2×A4 ρ10 3 3 -1 -1 3 -1 -1 -1 -1 -1 3 -1 0 0 0 0 orthogonal lifted from A4 ρ11 3 3 -1 -1 -1 3 -1 -1 -1 -1 -1 3 0 0 0 0 orthogonal lifted from A4 ρ12 3 -3 -1 -1 -1 -1 3 -3 1 1 1 1 0 0 0 0 orthogonal lifted from C2×A4 ρ13 3 3 3 -1 -1 -1 -1 -1 3 -1 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ14 3 3 -1 -1 -1 -1 3 3 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ15 3 -3 -1 -1 -1 3 -1 1 1 1 1 -3 0 0 0 0 orthogonal lifted from C2×A4 ρ16 3 -3 3 -1 -1 -1 -1 1 -3 1 1 1 0 0 0 0 orthogonal lifted from C2×A4

Permutation representations of C2×C22⋊A4
On 12 points - transitive group 12T56
Generators in S12
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(2 5)(3 6)(7 12)(9 11)
(1 4)(3 6)(7 12)(8 10)
(1 8)(2 5)(3 12)(4 10)(6 7)(9 11)
(1 10)(2 9)(3 6)(4 8)(5 11)(7 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)

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

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

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

G:=TransitiveGroup(12,56);

On 24 points - transitive group 24T176
Generators in S24
(1 8)(2 9)(3 7)(4 23)(5 24)(6 22)(10 20)(11 21)(12 19)(13 18)(14 16)(15 17)
(2 9)(3 7)(5 24)(6 22)(10 20)(11 21)(13 18)(15 17)
(1 8)(3 7)(4 23)(6 22)(11 21)(12 19)(13 18)(14 16)
(1 4)(2 20)(3 13)(5 15)(6 21)(7 18)(8 23)(9 10)(11 22)(12 16)(14 19)(17 24)
(1 14)(2 5)(3 21)(4 19)(6 13)(7 11)(8 16)(9 24)(10 17)(12 23)(15 20)(18 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)| (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (2,9)(3,7)(5,24)(6,22)(10,20)(11,21)(13,18)(15,17), (1,8)(3,7)(4,23)(6,22)(11,21)(12,19)(13,18)(14,16), (1,4)(2,20)(3,13)(5,15)(6,21)(7,18)(8,23)(9,10)(11,22)(12,16)(14,19)(17,24), (1,14)(2,5)(3,21)(4,19)(6,13)(7,11)(8,16)(9,24)(10,17)(12,23)(15,20)(18,22), (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( (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (2,9)(3,7)(5,24)(6,22)(10,20)(11,21)(13,18)(15,17), (1,8)(3,7)(4,23)(6,22)(11,21)(12,19)(13,18)(14,16), (1,4)(2,20)(3,13)(5,15)(6,21)(7,18)(8,23)(9,10)(11,22)(12,16)(14,19)(17,24), (1,14)(2,5)(3,21)(4,19)(6,13)(7,11)(8,16)(9,24)(10,17)(12,23)(15,20)(18,22), (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([[(1,8),(2,9),(3,7),(4,23),(5,24),(6,22),(10,20),(11,21),(12,19),(13,18),(14,16),(15,17)], [(2,9),(3,7),(5,24),(6,22),(10,20),(11,21),(13,18),(15,17)], [(1,8),(3,7),(4,23),(6,22),(11,21),(12,19),(13,18),(14,16)], [(1,4),(2,20),(3,13),(5,15),(6,21),(7,18),(8,23),(9,10),(11,22),(12,16),(14,19),(17,24)], [(1,14),(2,5),(3,21),(4,19),(6,13),(7,11),(8,16),(9,24),(10,17),(12,23),(15,20),(18,22)], [(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(24,176);

On 24 points - transitive group 24T177
Generators in S24
(1 12)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)
(1 12)(2 20)(3 7)(4 16)(5 15)(6 22)(8 19)(9 10)(11 21)(13 18)(14 23)(17 24)
(1 8)(2 10)(3 21)(4 23)(5 17)(6 13)(7 11)(9 20)(12 19)(14 16)(15 24)(18 22)
(1 16)(2 20)(3 22)(4 12)(5 15)(6 7)(8 14)(9 10)(11 13)(17 24)(18 21)(19 23)
(1 23)(2 17)(3 21)(4 8)(5 10)(6 13)(7 11)(9 15)(12 14)(16 19)(18 22)(20 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)

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

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

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

G:=TransitiveGroup(24,177);

On 24 points - transitive group 24T178
Generators in S24
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 10)(2 16)(3 14)(4 22)(5 9)(6 19)(7 24)(8 20)(11 13)(12 17)(15 18)(21 23)
(1 15)(2 11)(3 17)(4 20)(5 23)(6 7)(8 22)(9 21)(10 18)(12 14)(13 16)(19 24)
(1 15)(2 16)(3 12)(4 20)(5 9)(6 24)(7 19)(8 22)(10 18)(11 13)(14 17)(21 23)
(1 10)(2 13)(3 17)(4 22)(5 21)(6 7)(8 20)(9 23)(11 16)(12 14)(15 18)(19 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)

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

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

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

G:=TransitiveGroup(24,178);

C2×C22⋊A4 is a maximal subgroup of   C24⋊C12  C244Dic3  C2×A42
C2×C22⋊A4 is a maximal quotient of   C4○D4⋊A4  2+ 1+4.3C6

Polynomial with Galois group C2×C22⋊A4 over ℚ
actionf(x)Disc(f)
12T56x12-14x10+70x8-150x6+132x4-35x2+1212·54·138·532

Matrix representation of C2×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
,
 -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],[-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] >;

C2×C22⋊A4 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes A_4
% in TeX

G:=Group("C2xC2^2:A4");
// GroupNames label

G:=SmallGroup(96,229);
// by ID

G=gap.SmallGroup(96,229);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,116,225,730,1307]);
// Polycyclic

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

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