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G = C24⋊5A4  order 192 = 26·3

5th semidirect product of C24 and A4 acting faithfully

non-abelian, soluble, monomial

Aliases: C24⋊5A4, (C2×Q8)⋊2A4, C2.3(C23⋊A4), C24⋊C22⋊3C3, C22.8(C22⋊A4), SmallGroup(192,1024)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C24⋊C22 — C24⋊5A4
Chief series
C1 — C2 — C22 — C24 — C24⋊C22 — C24⋊5A4

Subgroups: 398 in 80 conjugacy classes, 12 normal (4 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×3], C22, C22 [×10], C6 [×3], C2×C4 [×3], D4 [×3], Q8 [×3], C23 [×8], A4 [×2], C2×C6, C42, C22⋊C4 [×6], C2×D4 [×3], C2×Q8 [×3], C24 [×2], SL2(𝔽3) [×3], C2×A4 [×6], C22≀C2 [×2], C4.4D4 [×3], C2×SL2(𝔽3) [×3], C22×A4 [×2], C24⋊C22, C24⋊5A4

Quotients:
C1, C3, A4 [×5], C22⋊A4, C23⋊A4 [×3], C24⋊5A4

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

Permutation representations
►On 16 points - transitive group 16T437
Generators in S16
(1 16)(2 8)(3 11)(4 5)(6 7)(9 10)(12 13)(14 15)

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

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

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

(5 11)(6 14)(7 10)(8 16)(9 12)(13 15)

(5 8)(6 12)(7 15)(9 14)(10 13)(11 16)

(5 6 7)(8 9 10)(11 12 13)(14 15 16)
 
G:=sub<Sym(16)| (1,16)(2,8)(3,11)(4,5)(6,7)(9,10)(12,13)(14,15), (1,14)(2,9)(3,12)(4,6)(5,7)(8,10)(11,13)(15,16), (1,4)(2,3)(5,16)(6,14)(7,15)(8,11)(9,12)(10,13), (1,2)(3,4)(5,11)(6,12)(7,13)(8,16)(9,14)(10,15), (5,11)(6,14)(7,10)(8,16)(9,12)(13,15), (5,8)(6,12)(7,15)(9,14)(10,13)(11,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16)>;
 
G:=Group( (1,16)(2,8)(3,11)(4,5)(6,7)(9,10)(12,13)(14,15), (1,14)(2,9)(3,12)(4,6)(5,7)(8,10)(11,13)(15,16), (1,4)(2,3)(5,16)(6,14)(7,15)(8,11)(9,12)(10,13), (1,2)(3,4)(5,11)(6,12)(7,13)(8,16)(9,14)(10,15), (5,11)(6,14)(7,10)(8,16)(9,12)(13,15), (5,8)(6,12)(7,15)(9,14)(10,13)(11,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16) );
 
G=PermutationGroup([(1,16),(2,8),(3,11),(4,5),(6,7),(9,10),(12,13),(14,15)], [(1,14),(2,9),(3,12),(4,6),(5,7),(8,10),(11,13),(15,16)], [(1,4),(2,3),(5,16),(6,14),(7,15),(8,11),(9,12),(10,13)], [(1,2),(3,4),(5,11),(6,12),(7,13),(8,16),(9,14),(10,15)], [(5,11),(6,14),(7,10),(8,16),(9,12),(13,15)], [(5,8),(6,12),(7,15),(9,14),(10,13),(11,16)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)])
 
G:=TransitiveGroup(16,437);
 

Matrix representation ►G ⊆ GL8(ℤ)

10000000
0-1000000
00100000
000-10000
00001000
00000-100
000000-10
00000001
,
10000000
01000000
00-100000
000-10000
00001000
00000-100
00000010
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010
,
10000000
00100000
00010000
01000000
00001000
00000010
00000001
00000100

G:=sub<GL(8,Integers())| [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,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;
 

Character table of C24⋊5A4

 class 12A2B2C2D2E3A3B4A4B4C6A6B6C6D6E6F
 size 111112121616121212161616161616
ρ111111111111111111    trivial
ρ2111111ζ3ζ32111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ3111111ζ32ζ3111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ43333-1-100-1-13000000    orthogonal lifted from A4
ρ53333-1-100-13-1000000    orthogonal lifted from A4
ρ63333-1300-1-1-1000000    orthogonal lifted from A4
ρ73333-1-1003-1-1000000    orthogonal lifted from A4
ρ833333-100-1-1-1000000    orthogonal lifted from A4
ρ94-44-400110001-1-1-11-1    orthogonal lifted from C23⋊A4
ρ1044-4-40011000-11-1-1-11    orthogonal lifted from C23⋊A4
ρ114-4-440011000-1-111-1-1    orthogonal lifted from C23⋊A4
ρ124-44-400ζ32ζ3000ζ3ζ65ζ65ζ6ζ32ζ6    complex lifted from C23⋊A4
ρ134-4-4400ζ3ζ32000ζ6ζ6ζ32ζ3ζ65ζ65    complex lifted from C23⋊A4
ρ144-4-4400ζ32ζ3000ζ65ζ65ζ3ζ32ζ6ζ6    complex lifted from C23⋊A4
ρ154-44-400ζ3ζ32000ζ32ζ6ζ6ζ65ζ3ζ65    complex lifted from C23⋊A4
ρ1644-4-400ζ3ζ32000ζ6ζ32ζ6ζ65ζ65ζ3    complex lifted from C23⋊A4
ρ1744-4-400ζ32ζ3000ζ65ζ3ζ65ζ6ζ6ζ32    complex lifted from C23⋊A4

In GAP, Magma, Sage, TeX 

C_2^4\rtimes_5A_4
 
% in TeX
 
G:=Group("C2^4:5A4");
 
// GroupNames label
 
G:=SmallGroup(192,1024);
 
// by ID
 
G=gap.SmallGroup(192,1024);
 
# by ID
 
G:=PCGroup([7,-3,-2,2,-2,2,-2,-2,85,191,675,1018,297,1264,1971,718]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^3=1,g*a*g^-1=a*b=b*a,a*c=c*a,e*a*e=a*d=d*a,f*a*f=a*c*d,e*b*e=b*c=c*b,f*b*f=b*d=d*b,g*b*g^-1=a,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
 
// generators/relations
 

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