C2^3.8S4 - GroupNames

G = C₂³.₈S₄ order 192 = 2⁶·3

2^nd non-split extension by C₂³ of S₄ acting via S₄/C₂²=S₃

non-abelian, soluble

Aliases: C₂³.₈S₄, C₂².GL₂(𝔽₃), C₂.C₄²⋊S₃, C₂.₃(C₄²⋊S₃), C₂³.₃A₄⋊₄C₂, SmallGroup(192,181)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₂.C₄² — C₂³.₃A₄ — C₂³.₈S₄

Chief series

C₁ — C₂ — C₂³ — C₂.C₄² — C₂³.₃A₄ — C₂³.₈S₄

Lower central

C₂³.₃A₄ — C₂³.₈S₄

Upper central

C₁ — C₂

Generators and relations for C₂³.₈S₄
G = < a,b,c,d,e,f,g | a²=b²=c²=f³=g²=1, d²=gag=fbf^-1=abc, e²=faf^-1=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, geg=be=eb, bg=gb, ede^-1=cd=dc, ce=ec, cf=fc, cg=gc, fdf^-1=cde, gdg=de, fef^-1=bcd, gfg=f^-1 >

C₁

C₂

₃C₂

₂₄C₂

₁₆C₃

C₂²

₃C₂²

₆C₄

₁₂C₂²

₁₂C₄

₁₂C₂²

₁₆S₃

₁₆C₆

₁₆S₃

C₂³

₃C₂×C₄

₆C₂³

₆C₂×C₄

₁₂D₄

₁₂C₈

₁₂C₂×C₄

₁₂D₄

₄A₄

₁₆D₆

₃C₄⋊C₄

₃C₂×D₄

₃C₂²×C₄

₆C₂²⋊C₄

₆C₂×C₈

₆C₂×D₄

₄S₄

₄C₂×A₄

C₂.C₄²

₃C₂²⋊C₈

₃C₄⋊D₄

₄C₂×S₄

₃C₂².SD₁₆

C₂³.₃A₄

C₂³.₈S₄

Character table of C₂³.₈S₄

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	6	8A	8B	8C	8D
size	1	1	3	3	24	32	6	6	12	24	32	12	12	12	12
ρ₁	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	linear of order 2
ρ₃	2	2	2	2	0	-1	2	2	2	0	-1	0	0	0	0	orthogonal lifted from S₃
ρ₄	2	-2	2	-2	0	-1	0	0	0	0	1	-√-2	√-2	√-2	-√-2	complex lifted from GL₂(𝔽₃)
ρ₅	2	-2	2	-2	0	-1	0	0	0	0	1	√-2	-√-2	-√-2	√-2	complex lifted from GL₂(𝔽₃)
ρ₆	3	3	3	3	1	0	-1	-1	-1	1	0	-1	-1	-1	-1	orthogonal lifted from S₄
ρ₇	3	3	3	3	-1	0	-1	-1	-1	-1	0	1	1	1	1	orthogonal lifted from S₄
ρ₈	3	3	-1	-1	1	0	-1+2i	-1-2i	1	-1	0	-i	-i	i	i	complex lifted from C₄²⋊S₃
ρ₉	3	3	-1	-1	1	0	-1-2i	-1+2i	1	-1	0	i	i	-i	-i	complex lifted from C₄²⋊S₃
ρ₁₀	3	3	-1	-1	-1	0	-1+2i	-1-2i	1	1	0	i	i	-i	-i	complex lifted from C₄²⋊S₃
ρ₁₁	3	3	-1	-1	-1	0	-1-2i	-1+2i	1	1	0	-i	-i	i	i	complex lifted from C₄²⋊S₃
ρ₁₂	4	-4	4	-4	0	1	0	0	0	0	-1	0	0	0	0	orthogonal lifted from GL₂(𝔽₃)
ρ₁₃	6	6	-2	-2	0	0	2	2	-2	0	0	0	0	0	0	orthogonal lifted from C₄²⋊S₃
ρ₁₄	6	-6	-2	2	0	0	0	0	0	0	0	√2	-√2	√2	-√2	orthogonal faithful
ρ₁₅	6	-6	-2	2	0	0	0	0	0	0	0	-√2	√2	-√2	√2	orthogonal faithful

Permutation representations of C₂³.₈S₄
►On 24 points - transitive group 24T313

Generators in S₂₄

(5 15)(6 16)(11 13)(12 14)

(1 8)(2 7)(3 9)(4 10)

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

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

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

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

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

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

G:=TransitiveGroup(24,313);

►On 24 points - transitive group 24T315

Generators in S₂₄

(3 4)(11 13)(12 14)(15 16)

(1 2)(5 6)(7 9)(8 10)

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

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

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

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

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

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

G:=TransitiveGroup(24,315);

►On 24 points - transitive group 24T426

Generators in S₂₄

(1 2)(3 5)(4 6)(7 8)(17 24)(18 21)(19 22)(20 23)

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

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

(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 4 8 3)(9 10 13 14)(11 12 15 16)(17 22)(19 24)

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

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

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

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

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

G:=TransitiveGroup(24,426);

►On 24 points - transitive group 24T428

Generators in S₂₄

(1 3)(2 4)(5 7)(6 8)(9 21)(10 22)(11 23)(12 24)

(1 2)(3 4)(5 8)(6 7)(13 20)(14 17)(15 18)(16 19)

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,428);

Matrix representation of C₂³.₈S₄ ►in GL₅(𝔽₇₃)

72	0	0	0	0
0	72	0	0	0
0	0	1	0	0
0	0	0	72	0
0	0	0	0	72

72	0	0	0	0
0	72	0	0	0
0	0	72	0	0
0	0	0	72	0
0	0	0	0	1

72	0	0	0	0
0	72	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

60	7	0	0	0
7	13	0	0	0
0	0	27	0	0
0	0	0	72	0
0	0	0	0	27

7	13	0	0	0
13	66	0	0	0
0	0	27	0	0
0	0	0	46	0
0	0	0	0	1

39	26	0	0	0
27	33	0	0	0
0	0	0	0	67
0	0	60	0	0
0	0	0	44	0

68	36	0	0	0
48	5	0	0	0
0	0	0	45	0
0	0	13	0	0
0	0	0	0	72

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1],[72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,7,0,0,0,7,13,0,0,0,0,0,27,0,0,0,0,0,72,0,0,0,0,0,27],[7,13,0,0,0,13,66,0,0,0,0,0,27,0,0,0,0,0,46,0,0,0,0,0,1],[39,27,0,0,0,26,33,0,0,0,0,0,0,60,0,0,0,0,0,44,0,0,67,0,0],[68,48,0,0,0,36,5,0,0,0,0,0,0,13,0,0,0,45,0,0,0,0,0,0,72] >;

C₂³.₈S₄ in GAP, Magma, Sage, TeX

C_2^3._8S_4

% in TeX

G:=Group("C2^3.8S4");

// GroupNames label

G:=SmallGroup(192,181);

// by ID

G=gap.SmallGroup(192,181);

# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,57,254,1143,268,171,934,521,80,2524,2531,3540]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂³.₈S₄ in TeX
Character table of C₂³.₈S₄ in TeX

G = C23.8S4 order 192 = 26·3

2nd non-split extension by C23 of S4 acting via S4/C22=S3

G = C₂³.₈S₄ order 192 = 2⁶·3

2^nd non-split extension by C₂³ of S₄ acting via S₄/C₂²=S₃