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## G = (C22×C4).A4order 192 = 26·3

### 4th 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 >

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

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