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G = (C22×C4).A4order 192 = 26·3

4th non-split extension by C22×C4 of A4 acting faithfully

non-abelian, soluble

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

C1C2C22×Q8 — (C22×C4).A4
C1C2C23C22×Q8Q8⋊A4 — (C22×C4).A4
C22×Q8 — (C22×C4).A4
C1C2

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 >

3C2
3C2
16C3
3C22
3C22
4C4
6C4
6C4
6C4
6C4
16C6
2Q8
2Q8
3C2×C4
3C2×C4
3C2×C4
3C2×C4
6C2×C4
6C2×C4
6C2×C4
6Q8
6C2×C4
6Q8
6C2×C4
4A4
16C12
3C22×C4
3C22×C4
6C2×Q8
6C4⋊C4
6C2×Q8
6C4⋊C4
4C2×A4
8SL2(𝔽3)
8SL2(𝔽3)
3C2.C42
3C2×C4⋊C4
4C4×A4

Character table of (C22×C4).A4

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B12A12B12C12D
 size 113316164412121212161616161616
ρ1111111111111111111    trivial
ρ2111111-1-1-111-111-1-1-1-1    linear of order 2
ρ31111ζ32ζ3-1-1-111-1ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ41111ζ3ζ32-1-1-111-1ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ51111ζ32ζ3111111ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ61111ζ3ζ32111111ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ72-2-22-1-12i-2i000011i-ii-i    complex lifted from C4.A4
ρ82-2-22-1-1-2i2i000011-ii-ii    complex lifted from C4.A4
ρ92-2-22ζ65ζ6-2i2i0000ζ32ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    complex lifted from C4.A4
ρ102-2-22ζ6ζ652i-2i0000ζ3ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    complex lifted from C4.A4
ρ112-2-22ζ6ζ65-2i2i0000ζ3ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    complex lifted from C4.A4
ρ122-2-22ζ65ζ62i-2i0000ζ32ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    complex lifted from C4.A4
ρ13333300-3-31-1-11000000    orthogonal lifted from C2×A4
ρ1433330033-1-1-1-1000000    orthogonal lifted from A4
ρ1566-2-200000-220000000    orthogonal lifted from C24⋊C6
ρ1666-2-2000002-20000000    orthogonal lifted from C24⋊C6
ρ176-62-20000200-2000000    symplectic faithful, Schur index 2
ρ186-62-20000-2002000000    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 23 7 21)(6 22 8 24)(9 12 11 10)(13 20 15 18)(14 19 16 17)
(1 11 3 9)(2 10 4 12)(5 8 7 6)(13 19 15 17)(14 18 16 20)(21 24 23 22)
(1 22 14)(2 23 15)(3 24 16)(4 21 13)(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)(5,23,7,21)(6,22,8,24)(9,12,11,10)(13,20,15,18)(14,19,16,17), (1,11,3,9)(2,10,4,12)(5,8,7,6)(13,19,15,17)(14,18,16,20)(21,24,23,22), (1,22,14)(2,23,15)(3,24,16)(4,21,13)(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)(5,23,7,21)(6,22,8,24)(9,12,11,10)(13,20,15,18)(14,19,16,17), (1,11,3,9)(2,10,4,12)(5,8,7,6)(13,19,15,17)(14,18,16,20)(21,24,23,22), (1,22,14)(2,23,15)(3,24,16)(4,21,13)(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),(5,23,7,21),(6,22,8,24),(9,12,11,10),(13,20,15,18),(14,19,16,17)], [(1,11,3,9),(2,10,4,12),(5,8,7,6),(13,19,15,17),(14,18,16,20),(21,24,23,22)], [(1,22,14),(2,23,15),(3,24,16),(4,21,13),(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)

121212
020000
201220
022222
120122
000002
,
201212
021212
221010
111220
002111
111022
,
001121
010001
021101
011201
112001
010002
,
121111
120110
111102
112211
210121
102201
,
212212
001211
220200
121210
020212
201101
,
012100
100120
210100
110200
120100
100201

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

Export

Subgroup lattice of (C22×C4).A4 in TeX
Character table of (C22×C4).A4 in TeX

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