C2^4.3A4 - GroupNames

G = C₂⁴.₃A₄ order 192 = 2⁶·3

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

Aliases: C₂⁴.₃A₄, C₂³.₃SL₂(𝔽₃), C₂³.₄Q₈⋊C₃, C₂³.₁₈(C₂×A₄), C₂.C₄²⋊₂C₆, C₂³.₃A₄⋊₃C₂, C₂.₃(C₂³.A₄), C₂².₄(C₂×SL₂(𝔽₃)), SmallGroup(192,198)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₂.C₄² — C₂⁴.₃A₄

Chief series

C₁ — C₂ — C₂³ — C₂.C₄² — C₂³.₃A₄ — C₂⁴.₃A₄

Lower central

C₂.C₄² — C₂⁴.₃A₄

Upper central

C₁ — C₂

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 >

C₁

C₂

₃C₂

₄C₂

₁₆C₃

C₂²

₃C₂²

₄C₂²

₆C₄

₆C₂²

₆C₄

₆C₂²

₁₂C₂²

₁₂C₄

₁₆C₆

C₂³

₃C₂×C₄

₆C₂³

₆C₂×C₄

₆C₂³

₆C₂×C₄

₄A₄

₁₆C₂×C₆

C₂⁴

₃C₂²×C₄

₆C₄⋊C₄

₆C₂²⋊C₄

₆C₄⋊C₄

₆C₂²⋊C₄

₄C₂×A₄

C₂.C₄²

₃C₂×C₂²⋊C₄

₃C₂×C₄⋊C₄

₄C₂²×A₄

C₂³.₄Q₈

C₂³.₃A₄

C₂⁴.₃A₄

Character table of C₂⁴.₃A₄

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F
size	1	1	3	3	4	4	16	16	12	12	12	12	16	16	16	16	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	1	1	1	1	-1	-1	-1	1	-1	-1	-1	1	linear of order 2
ρ₃	1	1	1	1	1	1	ζ₃	ζ₃²	1	1	1	1	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₄	1	1	1	1	-1	-1	ζ₃²	ζ₃	1	1	-1	-1	ζ₆⁵	ζ₃	ζ₆	ζ₆	ζ₆⁵	ζ₃²	linear of order 6
ρ₅	1	1	1	1	1	1	ζ₃²	ζ₃	1	1	1	1	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₆	1	1	1	1	-1	-1	ζ₃	ζ₃²	1	1	-1	-1	ζ₆	ζ₃²	ζ₆⁵	ζ₆⁵	ζ₆	ζ₃	linear of order 6
ρ₇	2	-2	2	-2	2	-2	-1	-1	0	0	0	0	-1	1	1	-1	1	1	symplectic lifted from SL₂(𝔽₃), Schur index 2
ρ₈	2	-2	2	-2	-2	2	-1	-1	0	0	0	0	1	1	-1	1	-1	1	symplectic lifted from SL₂(𝔽₃), Schur index 2
ρ₉	2	-2	2	-2	-2	2	ζ₆	ζ₆⁵	0	0	0	0	ζ₃	ζ₃	ζ₆	ζ₃²	ζ₆⁵	ζ₃²	complex lifted from SL₂(𝔽₃)
ρ₁₀	2	-2	2	-2	-2	2	ζ₆⁵	ζ₆	0	0	0	0	ζ₃²	ζ₃²	ζ₆⁵	ζ₃	ζ₆	ζ₃	complex lifted from SL₂(𝔽₃)
ρ₁₁	2	-2	2	-2	2	-2	ζ₆	ζ₆⁵	0	0	0	0	ζ₆⁵	ζ₃	ζ₃²	ζ₆	ζ₃	ζ₃²	complex lifted from SL₂(𝔽₃)
ρ₁₂	2	-2	2	-2	2	-2	ζ₆⁵	ζ₆	0	0	0	0	ζ₆	ζ₃²	ζ₃	ζ₆⁵	ζ₃²	ζ₃	complex lifted from SL₂(𝔽₃)
ρ₁₃	3	3	3	3	-3	-3	0	0	-1	-1	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₄	3	3	3	3	3	3	0	0	-1	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₅	6	6	-2	-2	0	0	0	0	-2	2	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³.A₄
ρ₁₆	6	6	-2	-2	0	0	0	0	2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³.A₄
ρ₁₇	6	-6	-2	2	0	0	0	0	0	0	2i	-2i	0	0	0	0	0	0	complex faithful
ρ₁₈	6	-6	-2	2	0	0	0	0	0	0	-2i	2i	0	0	0	0	0	0	complex faithful

Permutation representations of C₂⁴.₃A₄
►On 24 points - transitive group 24T293

Generators in S₂₄

(1 3)(6 8)(9 12)(14 16)(17 19)(18 20)

(1 3)(2 4)(5 7)(6 8)

(9 12)(10 11)(13 15)(14 16)

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

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

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

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

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

G:=TransitiveGroup(24,293);

►On 24 points - transitive group 24T307

Generators in S₂₄

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

(1 3)(2 4)(11 12)(13 14)

(5 16)(6 15)(7 9)(8 10)

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

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

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

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

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

G:=TransitiveGroup(24,307);

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

0	5	0	0	0	0
8	0	0	0	0	0
0	0	0	5	0	0
0	0	8	0	0	0
11	11	7	7	8	3
2	0	6	0	5	5

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	1	0
0	0	9	9	0	1

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
3	3	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	0	0
0	12	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
10	10	4	4	12	11
0	3	0	0	1	1

0	12	0	0	0	0
12	0	0	0	0	0
0	0	0	12	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	4	0	12	12

0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
10	10	4	4	12	11
1	0	0	0	0	0
0	0	0	0	0	9

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

G = C24.3A4 order 192 = 26·3

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

G = C₂⁴.₃A₄ order 192 = 2⁶·3

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