Copied to
clipboard

## G = C23.19(C2×A4)  order 192 = 26·3

### 12nd non-split extension by C23 of C2×A4 acting via C2×A4/C23=C3

Aliases: (C22×C4).5A4, C23.19(C2×A4), C22.4(C4.A4), C23.84C23⋊C3, C2.3(C42⋊C6), C23.3A4.1C2, C2.C42.1C6, SmallGroup(192,199)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2.C42 — C23.19(C2×A4)
 Chief series C1 — C2 — C23 — C2.C42 — C23.3A4 — C23.19(C2×A4)
 Lower central C2.C42 — C23.19(C2×A4)
 Upper central C1 — C2

Generators and relations for C23.19(C2×A4)
G = < a,b,c,d,e,f,g | a2=b2=c2=g3=1, d2=c, e2=a, f2=gbg-1=abc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, gag-1=b, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, ede-1=bcd, fdf-1=acd, dg=gd, geg-1=abcef, gfg-1=bce >

Character table of C23.19(C2×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 2i 0 0 -2i 0 0 0 0 0 0 complex faithful ρ16 6 -6 2 -2 0 0 0 0 -2i 0 0 2i 0 0 0 0 0 0 complex faithful ρ17 6 6 -2 -2 0 0 0 0 0 2i -2i 0 0 0 0 0 0 0 complex lifted from C42⋊C6 ρ18 6 6 -2 -2 0 0 0 0 0 -2i 2i 0 0 0 0 0 0 0 complex lifted from C42⋊C6

Permutation representations of C23.19(C2×A4)
On 24 points - transitive group 24T298
Generators in S24
(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 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)(2 10)(3 11)(4 12)(5 6)(7 8)(13 19 15 17)(14 18 16 20)(21 24)(22 23)
(1 4)(2 3)(5 24 7 22)(6 23 8 21)(9 10)(11 12)(13 20)(14 17)(15 18)(16 19)
(1 14 22)(2 15 23)(3 16 24)(4 13 21)(5 12 20)(6 9 17)(7 10 18)(8 11 19)

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

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

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

G:=TransitiveGroup(24,298);

On 24 points - transitive group 24T310
Generators in S24
(5 7)(6 8)(21 23)(22 24)
(1 3)(2 4)(9 11)(10 12)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 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)(2 10)(3 11)(4 12)(5 22 7 24)(6 21 8 23)(13 15)(18 20)
(1 3)(5 24)(6 21)(7 22)(8 23)(10 12)(13 17 15 19)(14 20 16 18)
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 21 15)(10 22 16)(11 23 13)(12 24 14)

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

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

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

G:=TransitiveGroup(24,310);

On 24 points - transitive group 24T312
Generators in S24
(1 3)(2 4)(9 11)(10 12)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 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 3 11)(2 12 4 10)(13 15)(14 16)(17 21)(18 24)(19 23)(20 22)
(5 16)(6 15)(7 14)(8 13)(9 11)(10 12)(17 21 19 23)(18 24 20 22)
(1 19 6)(2 20 7)(3 17 8)(4 18 5)(9 21 15)(10 22 16)(11 23 13)(12 24 14)

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

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

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

G:=TransitiveGroup(24,312);

Matrix representation of C23.19(C2×A4) in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0
,
 6 2 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8
,
 1 6 0 0 0 0 0 0 4 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0
,
 5 6 0 0 0 0 0 0 3 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0],[6,1,0,0,0,0,0,0,2,7,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8],[1,4,0,0,0,0,0,0,6,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0],[5,3,0,0,0,0,0,0,6,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0] >;

C23.19(C2×A4) in GAP, Magma, Sage, TeX

C_2^3._{19}(C_2\times A_4)
% in TeX

G:=Group("C2^3.19(C2xA4)");
// GroupNames label

G:=SmallGroup(192,199);
// by ID

G=gap.SmallGroup(192,199);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,632,135,268,4371,934,521,304,2531,1524]);
// Polycyclic

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

Export

׿
×
𝔽