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G = C24.3A4order 192 = 26·3

3rd non-split extension by C24 of A4 acting faithfully

non-abelian, soluble

Aliases: C24.3A4, C23.3SL2(𝔽3), C23.4Q8⋊C3, C23.18(C2×A4), C2.C422C6, C23.3A43C2, C2.3(C23.A4), C22.4(C2×SL2(𝔽3)), SmallGroup(192,198)

Series: Derived Chief Lower central Upper central

C1C2C2.C42 — C24.3A4
C1C2C23C2.C42C23.3A4 — C24.3A4
C2.C42 — C24.3A4
C1C2

Generators and relations for C24.3A4
 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, eae-1=abc, faf-1=abd, 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 >

3C2
3C2
4C2
4C2
16C3
3C22
3C22
4C22
6C4
6C22
6C4
6C22
12C22
12C4
16C6
16C6
16C6
3C2×C4
3C2×C4
6C23
6C2×C4
6C23
6C2×C4
6C2×C4
6C2×C4
6C2×C4
4A4
16C2×C6
3C22×C4
3C22×C4
6C4⋊C4
6C22⋊C4
6C4⋊C4
6C22⋊C4
4C2×A4
4C2×A4
4C2×A4
3C2×C22⋊C4
3C2×C4⋊C4
4C22×A4

Character table of C24.3A4

 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 C23.A4
ρ1666-2-200002-200000000    orthogonal lifted from C23.A4
ρ176-6-220000002i-2i000000    complex faithful
ρ186-6-22000000-2i2i000000    complex faithful

Permutation representations of C24.3A4
On 24 points - transitive group 24T293
Generators in S24
(2 14)(4 7)(5 12)(10 15)(17 19)(18 20)
(3 8)(4 7)(5 12)(6 11)
(1 13)(2 14)(9 16)(10 15)
(1 13)(2 14)(3 8)(4 7)(5 12)(6 11)(9 16)(10 15)(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 10)(2 16)(3 11 8 6)(4 5 7 12)(9 14)(13 15)(17 23)(18 22)(19 21)(20 24)
(1 24 11)(2 20 7)(3 16 21)(4 14 18)(5 10 17)(6 13 22)(8 9 23)(12 15 19)

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

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

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

G:=TransitiveGroup(24,293);

On 24 points - transitive group 24T307
Generators in S24
(1 13)(2 12)(3 14)(4 11)(5 9)(6 15)(7 16)(8 10)(17 22)(18 23)(19 24)(20 21)
(1 3)(2 4)(11 12)(13 14)
(5 7)(6 8)(9 16)(10 15)
(1 3)(2 4)(5 7)(6 8)(9 16)(10 15)(11 12)(13 14)(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 12 3 11)(2 14 4 13)(5 7)(10 15)(17 20)(18 19)(21 24)(22 23)
(1 7 18)(2 15 22)(3 5 20)(4 10 24)(6 17 12)(8 19 11)(9 21 14)(13 16 23)

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

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

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

G:=TransitiveGroup(24,307);

Matrix representation of C24.3A4 in GL6(𝔽13)

050000
800000
000500
008000
11117783
206055
,
100000
010000
0012000
0001200
000010
009901
,
1200000
0120000
001000
000100
000010
330001
,
1200000
0120000
0012000
0001200
0000120
0000012
,
100000
0120000
000100
001000
1010441211
030011
,
0120000
1200000
0001200
001000
000010
00401212
,
001000
000100
000010
1010441211
100000
000009

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

C24.3A4 in GAP, Magma, Sage, TeX

C_2^4._3A_4
% in TeX

G:=Group("C2^4.3A4");
// GroupNames label

G:=SmallGroup(192,198);
// by ID

G=gap.SmallGroup(192,198);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,1640,135,604,1011,934,521,304,851,1524]);
// Polycyclic

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

Export

Subgroup lattice of C24.3A4 in TeX
Character table of C24.3A4 in TeX

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