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## G = C2×C23.3A4order 192 = 26·3

### Direct product of C2 and C23.3A4

Aliases: C2×C23.3A4, C24.8A4, C23.4SL2(𝔽3), C23.9(C2×A4), C2.C423C6, C22.2(C42⋊C3), C22.1(C2×SL2(𝔽3)), C2.2(C2×C42⋊C3), (C2×C2.C42)⋊C3, SmallGroup(192,189)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2.C42 — C2×C23.3A4
 Chief series C1 — C2 — C23 — C2.C42 — C23.3A4 — C2×C23.3A4
 Lower central C2.C42 — C2×C23.3A4
 Upper central C1 — C22

Generators and relations for C2×C23.3A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=gbg-1=bcd, f2=gcg-1=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef-1=de=ed, df=fd, dg=gd, geg-1=bef, gfg-1=cde >

Subgroups: 327 in 75 conjugacy classes, 14 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C22 [×2], C22 [×11], C6 [×3], C2×C4 [×16], C23, C23 [×2], C23 [×4], A4, C2×C6, C22×C4 [×10], C24, C2×A4 [×3], C2.C42, C2.C42, C23×C4, C22×A4, C2×C2.C42, C23.3A4, C2×C23.3A4
Quotients: C1, C2, C3, C6, A4, SL2(𝔽3) [×2], C2×A4, C42⋊C3, C2×SL2(𝔽3), C23.3A4 [×2], C2×C42⋊C3, C2×C23.3A4

Character table of C2×C23.3A4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F size 1 1 1 1 3 3 3 3 16 16 6 6 6 6 6 6 6 6 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 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 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 1 -1 1 -1 -1 1 1 -1 ζ32 ζ3 1 1 1 1 -1 -1 -1 -1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ5 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 -1 1 -1 -1 1 1 -1 ζ3 ζ32 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ7 2 -2 -2 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 -1 1 -1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ8 2 2 -2 -2 2 -2 2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 -1 1 -1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ9 2 2 -2 -2 2 -2 2 -2 ζ6 ζ65 0 0 0 0 0 0 0 0 ζ3 ζ32 ζ65 ζ3 ζ6 ζ32 complex lifted from SL2(𝔽3) ρ10 2 2 -2 -2 2 -2 2 -2 ζ65 ζ6 0 0 0 0 0 0 0 0 ζ32 ζ3 ζ6 ζ32 ζ65 ζ3 complex lifted from SL2(𝔽3) ρ11 2 -2 -2 2 -2 -2 2 2 ζ65 ζ6 0 0 0 0 0 0 0 0 ζ32 ζ3 ζ32 ζ6 ζ3 ζ65 complex lifted from SL2(𝔽3) ρ12 2 -2 -2 2 -2 -2 2 2 ζ6 ζ65 0 0 0 0 0 0 0 0 ζ3 ζ32 ζ3 ζ65 ζ32 ζ6 complex lifted from SL2(𝔽3) ρ13 3 -3 3 -3 -3 3 3 -3 0 0 -1 -1 -1 -1 1 1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 3 3 3 3 3 3 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 3 -3 3 -3 1 -1 -1 1 0 0 1 -1-2i -1+2i 1 -1 1+2i 1-2i -1 0 0 0 0 0 0 complex lifted from C2×C42⋊C3 ρ16 3 3 3 3 -1 -1 -1 -1 0 0 1 -1-2i -1+2i 1 1 -1-2i -1+2i 1 0 0 0 0 0 0 complex lifted from C42⋊C3 ρ17 3 -3 3 -3 1 -1 -1 1 0 0 -1-2i 1 1 -1+2i 1+2i -1 -1 1-2i 0 0 0 0 0 0 complex lifted from C2×C42⋊C3 ρ18 3 3 3 3 -1 -1 -1 -1 0 0 1 -1+2i -1-2i 1 1 -1+2i -1-2i 1 0 0 0 0 0 0 complex lifted from C42⋊C3 ρ19 3 3 3 3 -1 -1 -1 -1 0 0 -1+2i 1 1 -1-2i -1+2i 1 1 -1-2i 0 0 0 0 0 0 complex lifted from C42⋊C3 ρ20 3 3 3 3 -1 -1 -1 -1 0 0 -1-2i 1 1 -1+2i -1-2i 1 1 -1+2i 0 0 0 0 0 0 complex lifted from C42⋊C3 ρ21 3 -3 3 -3 1 -1 -1 1 0 0 1 -1+2i -1-2i 1 -1 1-2i 1+2i -1 0 0 0 0 0 0 complex lifted from C2×C42⋊C3 ρ22 3 -3 3 -3 1 -1 -1 1 0 0 -1+2i 1 1 -1-2i 1-2i -1 -1 1+2i 0 0 0 0 0 0 complex lifted from C2×C42⋊C3 ρ23 6 6 -6 -6 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23.3A4 ρ24 6 -6 -6 6 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23.3A4

Permutation representations of C2×C23.3A4
On 24 points - transitive group 24T422
Generators in S24
(1 7)(2 8)(3 4)(5 6)(9 11)(10 12)(13 22)(14 23)(15 24)(16 21)(17 19)(18 20)
(1 7)(2 8)(3 4)(5 6)(9 20)(10 17)(11 18)(12 19)
(1 2)(3 6)(4 5)(7 8)(13 22)(14 23)(15 24)(16 21)
(1 8)(2 7)(3 5)(4 6)(9 18)(10 19)(11 20)(12 17)(13 24)(14 21)(15 22)(16 23)
(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 6 7 5)(2 3 8 4)(9 12 20 19)(10 18 17 11)(14 21)(16 23)
(1 14 12)(2 16 19)(3 24 11)(4 15 9)(5 13 20)(6 22 18)(7 23 10)(8 21 17)

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

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

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

G:=TransitiveGroup(24,422);

On 24 points - transitive group 24T423
Generators in S24
(1 10)(2 9)(3 5)(4 6)(7 8)(11 16)(12 15)(13 14)(17 24)(18 21)(19 22)(20 23)
(1 11)(2 12)(9 15)(10 16)
(3 6)(4 5)(7 14)(8 13)
(1 11)(2 12)(3 6)(4 5)(7 14)(8 13)(9 15)(10 16)(17 19)(18 20)(21 23)(22 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)
(1 9 11 15)(2 16 12 10)(3 14)(4 8)(5 13)(6 7)(17 24)(18 23)(19 22)(20 21)
(1 7 19)(2 3 21)(4 20 15)(5 18 9)(6 23 12)(8 22 10)(11 14 17)(13 24 16)

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

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

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

G:=TransitiveGroup(24,423);

On 24 points - transitive group 24T424
Generators in S24
(1 4)(2 3)(5 11)(6 12)(7 16)(8 15)(9 13)(10 14)(17 24)(18 21)(19 22)(20 23)
(1 2)(3 4)(5 6)(11 12)
(7 14)(8 13)(9 15)(10 16)
(1 2)(3 4)(5 6)(7 14)(8 13)(9 15)(10 16)(11 12)(17 19)(18 20)(21 23)(22 24)
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 5 2 6)(3 12 4 11)(7 14)(10 16)(17 20)(18 19)(21 22)(23 24)
(1 14 20)(2 7 18)(3 16 21)(4 10 23)(5 8 19)(6 13 17)(9 24 12)(11 15 22)

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

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

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

G:=TransitiveGroup(24,424);

Matrix representation of C2×C23.3A4 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 8
,
 9 10 0 0 0 10 4 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 12
,
 3 12 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,8],[9,10,0,0,0,10,4,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,12],[3,0,0,0,0,12,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C2×C23.3A4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._3A_4
% in TeX

G:=Group("C2xC2^3.3A4");
// GroupNames label

G:=SmallGroup(192,189);
// by ID

G=gap.SmallGroup(192,189);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,135,268,934,521,80,2531,3540]);
// Polycyclic

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

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