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G = C24.7A4  order 192 = 26·3

7th non-split extension by C24 of A4 acting faithfully

non-abelian, soluble

Aliases: C24.7A4, C23⋊2SL2(𝔽3), (C2×Q8)⋊1A4, C23⋊2Q8⋊C3, C2.1(Q8⋊A4), C2.1(C23⋊A4), C22.6(C22⋊A4), SmallGroup(192,1021)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C23⋊2Q8 — C24.7A4
Chief series
C1 — C2 — C22 — C2×Q8 — C23⋊2Q8 — C24.7A4

Subgroups: 326 in 72 conjugacy classes, 13 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C22, C22 [×6], C6 [×3], C2×C4 [×6], Q8 [×4], C23, C23 [×4], A4, C2×C6, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C2×Q8 [×4], C24, SL2(𝔽3) [×4], C2×A4 [×3], C2×C22⋊C4, C22⋊Q8 [×4], C2×SL2(𝔽3) [×4], C22×A4, C23⋊2Q8, C24.7A4

Quotients:
C1, C3, A4 [×5], SL2(𝔽3), C22⋊A4, Q8⋊A4, C23⋊A4 [×2], C24.7A4

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=f2=d, gag-1=ab=ba, faf-1=ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, bf=fb, gbg-1=a, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, geg-1=def, gfg-1=e >

Permutation representations
►On 16 points - transitive group 16T438
Generators in S16
(5 11)(6 12)(7 9)(8 10)

(2 14)(4 16)(5 11)(7 9)

(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 8 3 6)(2 7 4 5)(9 16 11 14)(10 15 12 13)

(2 8 5)(4 6 7)(9 16 12)(10 11 14)
 
G:=sub<Sym(16)| (5,11)(6,12)(7,9)(8,10), (2,14)(4,16)(5,11)(7,9), (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,3,6)(2,7,4,5)(9,16,11,14)(10,15,12,13), (2,8,5)(4,6,7)(9,16,12)(10,11,14)>;
 
G:=Group( (5,11)(6,12)(7,9)(8,10), (2,14)(4,16)(5,11)(7,9), (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,3,6)(2,7,4,5)(9,16,11,14)(10,15,12,13), (2,8,5)(4,6,7)(9,16,12)(10,11,14) );
 
G=PermutationGroup([(5,11),(6,12),(7,9),(8,10)], [(2,14),(4,16),(5,11),(7,9)], [(1,13),(2,14),(3,15),(4,16),(5,11),(6,12),(7,9),(8,10)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8,3,6),(2,7,4,5),(9,16,11,14),(10,15,12,13)], [(2,8,5),(4,6,7),(9,16,12),(10,11,14)])
 
G:=TransitiveGroup(16,438);
 

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
001000
0001200
000010
001201212
,
100000
010000
001000
000100
0000120
001212012
,
100000
010000
0012000
0001200
0000120
0000012
,
1200000
0120000
0012000
0001200
0000120
0000012
,
340000
4100000
0000120
001112
001000
001212012
,
0120000
100000
0001200
001000
0012121211
000111
,
100000
1090000
001000
000010
0012121211
000001

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

Character table of C24.7A4

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D6E6F
 size 111166161612121212161616161616
ρ1111111111111111111    trivial
ρ2111111ζ3ζ321111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ3111111ζ32ζ31111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ42-22-22-2-1-10000-1111-11    symplectic lifted from SL2(𝔽3), Schur index 2
ρ52-22-22-2ζ6ζ650000ζ65ζ3ζ3ζ32ζ6ζ32    complex lifted from SL2(𝔽3)
ρ62-22-22-2ζ65ζ60000ζ6ζ32ζ32ζ3ζ65ζ3    complex lifted from SL2(𝔽3)
ρ73333-1-100-1-1-13000000    orthogonal lifted from A4
ρ83333-1-100-1-13-1000000    orthogonal lifted from A4
ρ933333300-1-1-1-1000000    orthogonal lifted from A4
ρ103333-1-1003-1-1-1000000    orthogonal lifted from A4
ρ113333-1-100-13-1-1000000    orthogonal lifted from A4
ρ1244-4-400110000-11-1-1-11    orthogonal lifted from C23⋊A4
ρ134-4-4400110000-1-111-1-1    orthogonal lifted from C23⋊A4
ρ144-4-4400ζ32ζ30000ζ65ζ65ζ3ζ32ζ6ζ6    complex lifted from C23⋊A4
ρ154-4-4400ζ3ζ320000ζ6ζ6ζ32ζ3ζ65ζ65    complex lifted from C23⋊A4
ρ1644-4-400ζ3ζ320000ζ6ζ32ζ6ζ65ζ65ζ3    complex lifted from C23⋊A4
ρ1744-4-400ζ32ζ30000ζ65ζ3ζ65ζ6ζ6ζ32    complex lifted from C23⋊A4
ρ186-66-6-22000000000000    symplectic lifted from Q8⋊A4, Schur index 2

In GAP, Magma, Sage, TeX 

C_2^4._7A_4
 
% in TeX
 
G:=Group("C2^4.7A4");
 
// GroupNames label
 
G:=SmallGroup(192,1021);
 
// by ID
 
G=gap.SmallGroup(192,1021);
 
# by ID
 
G:=PCGroup([7,-3,-2,2,-2,2,-2,-2,85,191,675,297,248,1264,851,375,172]);
 
// 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,g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,g*b*g^-1=a,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,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
 

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