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G = C24.A4order 192 = 26·3

1st non-split extension by C24 of A4 acting faithfully

non-abelian, soluble

Aliases: C24.1A4, C231SL2(𝔽3), C23⋊Q8⋊C3, Q8⋊A41C2, (C22×Q8)⋊1C6, C23.15(C2×A4), C2.2(C24⋊C6), C22.2(C2×SL2(𝔽3)), SmallGroup(192,195)

Series: Derived Chief Lower central Upper central

C1C2C22×Q8 — C24.A4
C1C2C23C22×Q8Q8⋊A4 — C24.A4
C22×Q8 — C24.A4
C1C2

Generators and relations for C24.A4
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=f2=d, eae-1=ab=ba, faf-1=ac=ca, ad=da, ag=ga, gbg-1=bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, gcg-1=b, fef-1=de=ed, df=fd, dg=gd, geg-1=def, gfg-1=e >

Subgroups: 299 in 57 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×6], C6 [×3], C2×C4 [×7], Q8 [×4], C23, C23 [×2], C23 [×2], A4, C2×C6, C22⋊C4 [×2], C22×C4 [×2], C2×Q8 [×2], C24, SL2(𝔽3) [×2], C2×A4 [×3], C2.C42, C2×C22⋊C4, C22×Q8, C22×A4, C23⋊Q8, Q8⋊A4, C24.A4
Quotients: C1, C2, C3, C6, A4, SL2(𝔽3) [×2], C2×A4, C2×SL2(𝔽3), C24⋊C6, C24.A4

Character table of C24.A4

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D6E6F
 size 113344161612121212161616161616
ρ1111111111111111111    trivial
ρ21111-1-11111-1-1-11-1-1-11    linear of order 2
ρ3111111ζ3ζ321111ζ32ζ32ζ3ζ3ζ32ζ3    linear of order 3
ρ41111-1-1ζ32ζ311-1-1ζ65ζ3ζ6ζ6ζ65ζ32    linear of order 6
ρ5111111ζ32ζ31111ζ3ζ3ζ32ζ32ζ3ζ32    linear of order 3
ρ61111-1-1ζ3ζ3211-1-1ζ6ζ32ζ65ζ65ζ6ζ3    linear of order 6
ρ72-22-22-2-1-10000-111-111    symplectic lifted from SL2(𝔽3), Schur index 2
ρ82-22-2-22-1-1000011-11-11    symplectic lifted from SL2(𝔽3), Schur index 2
ρ92-22-2-22ζ6ζ650000ζ3ζ3ζ6ζ32ζ65ζ32    complex lifted from SL2(𝔽3)
ρ102-22-2-22ζ65ζ60000ζ32ζ32ζ65ζ3ζ6ζ3    complex lifted from SL2(𝔽3)
ρ112-22-22-2ζ6ζ650000ζ65ζ3ζ32ζ6ζ3ζ32    complex lifted from SL2(𝔽3)
ρ122-22-22-2ζ65ζ60000ζ6ζ32ζ3ζ65ζ32ζ3    complex lifted from SL2(𝔽3)
ρ133333-3-300-1-111000000    orthogonal lifted from C2×A4
ρ1433333300-1-1-1-1000000    orthogonal lifted from A4
ρ1566-2-20000-2200000000    orthogonal lifted from C24⋊C6
ρ1666-2-200002-200000000    orthogonal lifted from C24⋊C6
ρ176-6-220000002i-2i000000    complex faithful
ρ186-6-22000000-2i2i000000    complex faithful

Permutation representations of C24.A4
On 24 points - transitive group 24T291
Generators in S24
(2 4)(5 7)(10 12)(17 19)(18 20)(21 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(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)(5 22 7 24)(6 21 8 23)(13 19 15 17)(14 18 16 20)
(1 6 13)(2 21 20)(3 8 15)(4 23 18)(5 17 12)(7 19 10)(9 22 14)(11 24 16)

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

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

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

G:=TransitiveGroup(24,291);

On 24 points - transitive group 24T303
Generators in S24
(1 4)(2 3)(5 24)(6 23)(7 22)(8 21)(9 10)(11 12)(13 18)(14 19)(15 20)(16 17)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(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)(5 22 7 24)(6 21 8 23)(13 19 15 17)(14 18 16 20)
(1 6 13)(2 21 20)(3 8 15)(4 23 18)(5 17 12)(7 19 10)(9 22 14)(11 24 16)

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

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

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

G:=TransitiveGroup(24,303);

Matrix representation of C24.A4 in GL6(𝔽13)

1220000
010000
0901200
0912000
0300012
0300120
,
1200000
0120000
0012000
0001200
1000010
1000001
,
1200000
0120000
401000
400100
0000120
0000012
,
1200000
0120000
0012000
0001200
0000120
0000012
,
1220000
1210000
598000
1290500
630008
630080
,
800000
850000
000800
008000
7000012
400010
,
9011000
0012100
004010
004001
0010000
0110000

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,2,1,9,9,3,3,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,10,10,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,4,4,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,12,5,12,6,6,2,1,9,9,3,3,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[8,8,0,0,7,4,0,5,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[9,0,0,0,0,0,0,0,0,0,0,1,11,12,4,4,10,10,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C24.A4 in GAP, Magma, Sage, TeX

C_2^4.A_4
% in TeX

G:=Group("C2^4.A4");
// GroupNames label

G:=SmallGroup(192,195);
// by ID

G=gap.SmallGroup(192,195);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,632,135,352,1683,262,521,248,851,1524]);
// Polycyclic

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

Export

Character table of C24.A4 in TeX

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