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G = C2×C23.3A4order 192 = 26·3

Direct product of C2 and C23.3A4

direct product, non-abelian, soluble

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

C1C2C2.C42 — C2×C23.3A4
C1C2C23C2.C42C23.3A4 — C2×C23.3A4
C2.C42 — C2×C23.3A4
C1C22

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 12A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H6A6B6C6D6E6F
 size 11113333161666666666161616161616
ρ1111111111111111111111111    trivial
ρ21-11-1-111-1111111-1-1-1-111-1-1-1-1    linear of order 2
ρ311111111ζ32ζ311111111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ41-11-1-111-1ζ32ζ31111-1-1-1-1ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ511111111ζ3ζ3211111111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ61-11-1-111-1ζ3ζ321111-1-1-1-1ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ72-2-22-2-222-1-100000000111-11-1    symplectic lifted from SL2(𝔽3), Schur index 2
ρ822-2-22-22-2-1-10000000011-11-11    symplectic lifted from SL2(𝔽3), Schur index 2
ρ922-2-22-22-2ζ6ζ6500000000ζ3ζ32ζ65ζ3ζ6ζ32    complex lifted from SL2(𝔽3)
ρ1022-2-22-22-2ζ65ζ600000000ζ32ζ3ζ6ζ32ζ65ζ3    complex lifted from SL2(𝔽3)
ρ112-2-22-2-222ζ65ζ600000000ζ32ζ3ζ32ζ6ζ3ζ65    complex lifted from SL2(𝔽3)
ρ122-2-22-2-222ζ6ζ6500000000ζ3ζ32ζ3ζ65ζ32ζ6    complex lifted from SL2(𝔽3)
ρ133-33-3-333-300-1-1-1-11111000000    orthogonal lifted from C2×A4
ρ143333333300-1-1-1-1-1-1-1-1000000    orthogonal lifted from A4
ρ153-33-31-1-11001-1-2i-1+2i1-11+2i1-2i-1000000    complex lifted from C2×C42⋊C3
ρ163333-1-1-1-1001-1-2i-1+2i11-1-2i-1+2i1000000    complex lifted from C42⋊C3
ρ173-33-31-1-1100-1-2i11-1+2i1+2i-1-11-2i000000    complex lifted from C2×C42⋊C3
ρ183333-1-1-1-1001-1+2i-1-2i11-1+2i-1-2i1000000    complex lifted from C42⋊C3
ρ193333-1-1-1-100-1+2i11-1-2i-1+2i11-1-2i000000    complex lifted from C42⋊C3
ρ203333-1-1-1-100-1-2i11-1+2i-1-2i11-1+2i000000    complex lifted from C42⋊C3
ρ213-33-31-1-11001-1+2i-1-2i1-11-2i1+2i-1000000    complex lifted from C2×C42⋊C3
ρ223-33-31-1-1100-1+2i11-1-2i1-2i-1-11+2i000000    complex lifted from C2×C42⋊C3
ρ2366-6-6-22-220000000000000000    orthogonal lifted from C23.3A4
ρ246-6-6622-2-20000000000000000    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)

120000
012000
001200
000120
000012
,
120000
012000
001200
000120
00001
,
120000
012000
001200
00010
000012
,
120000
012000
00100
00010
00001
,
01000
120000
00100
00050
00008
,
910000
104000
00500
00050
000012
,
312000
09000
00010
00001
00100

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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Character table of C2×C23.3A4 in TeX

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