Copied to
clipboard

G = C23.8S4order 192 = 26·3

2nd 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), C23.3A44C2, SmallGroup(192,181)

Series: Derived Chief Lower central Upper central

C1C2C2.C42C23.3A4 — C23.8S4
C1C2C23C2.C42C23.3A4 — C23.8S4
C23.3A4 — C23.8S4
C1C2

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 >

3C2
3C2
24C2
16C3
3C22
3C22
6C4
6C4
12C22
12C4
12C22
12C22
16S3
16C6
16S3
3C2×C4
3C2×C4
6C23
6C2×C4
12D4
12D4
12C8
12C2×C4
12D4
4A4
16D6
3C4⋊C4
3C2×D4
3C22×C4
6C22⋊C4
6C2×C8
6C2×D4
4S4
4S4
4C2×A4
3C22⋊C8
3C4⋊D4
4C2×S4
3C22.SD16

Character table of C23.8S4

 class 12A2B2C2D34A4B4C4D68A8B8C8D
 size 113324326612243212121212
ρ1111111111111111    trivial
ρ21111-11111-11-1-1-1-1    linear of order 2
ρ322220-12220-10000    orthogonal lifted from S3
ρ42-22-20-100001--2-2-2--2    complex lifted from GL2(𝔽3)
ρ52-22-20-100001-2--2--2-2    complex lifted from GL2(𝔽3)
ρ6333310-1-1-110-1-1-1-1    orthogonal lifted from S4
ρ73333-10-1-1-1-101111    orthogonal lifted from S4
ρ833-1-110-1+2i-1-2i1-10-i-iii    complex lifted from C42⋊S3
ρ933-1-110-1-2i-1+2i1-10ii-i-i    complex lifted from C42⋊S3
ρ1033-1-1-10-1+2i-1-2i110ii-i-i    complex lifted from C42⋊S3
ρ1133-1-1-10-1-2i-1+2i110-i-iii    complex lifted from C42⋊S3
ρ124-44-4010000-10000    orthogonal lifted from GL2(𝔽3)
ρ1366-2-20022-2000000    orthogonal lifted from C42⋊S3
ρ146-6-2200000002-22-2    orthogonal faithful
ρ156-6-220000000-22-22    orthogonal faithful

Permutation representations of C23.8S4
On 24 points - transitive group 24T313
Generators in S24
(9 15)(10 16)(11 13)(12 14)
(1 4)(2 3)(5 7)(6 8)
(1 4)(2 3)(5 7)(6 8)(9 15)(10 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 6 4 8)(2 7 3 5)(9 11)(10 14)(12 16)(13 15)(17 24)(18 23)(19 22)(20 21)
(1 12 17)(2 16 21)(3 10 23)(4 14 19)(5 11 22)(6 9 20)(7 13 24)(8 15 18)
(2 7)(3 5)(6 8)(9 18)(10 22)(11 23)(12 17)(13 21)(14 19)(15 20)(16 24)

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

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

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

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

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

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

G:=TransitiveGroup(24,315);

On 24 points - transitive group 24T426
Generators in S24
(1 7)(2 8)(3 4)(5 6)(9 22)(10 23)(11 24)(12 21)
(1 2)(3 5)(4 6)(7 8)(13 19)(14 20)(15 17)(16 18)
(1 8)(2 7)(3 6)(4 5)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)
(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 4 2 6)(3 8 5 7)(9 24)(11 22)(13 18 19 16)(14 15 20 17)
(1 24 16)(2 11 14)(3 21 17)(4 23 15)(5 12 19)(6 10 13)(7 22 18)(8 9 20)
(1 16)(2 18)(3 17)(4 19)(5 15)(6 13)(7 14)(8 20)(11 22)(12 23)

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

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

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

G:=TransitiveGroup(24,426);

On 24 points - transitive group 24T428
Generators in S24
(1 3)(2 4)(5 7)(6 8)(13 21)(14 22)(15 23)(16 24)
(1 2)(3 4)(5 8)(6 7)(9 19)(10 20)(11 17)(12 18)
(1 4)(2 3)(5 6)(7 8)(9 17)(10 18)(11 19)(12 20)(13 23)(14 24)(15 21)(16 22)
(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 12 19 18)(10 17 20 11)(13 21)(14 16)(15 23)(22 24)
(1 22 11)(2 14 9)(3 24 17)(4 16 19)(5 15 20)(6 21 12)(7 13 10)(8 23 18)
(1 10)(2 20)(3 12)(4 18)(5 9)(6 17)(7 11)(8 19)(13 22)(14 15)(16 23)(21 24)

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

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

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

G:=TransitiveGroup(24,428);

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

720000
072000
00100
000720
000072
,
720000
072000
007200
000720
00001
,
720000
072000
00100
00010
00001
,
607000
713000
002700
000720
000027
,
713000
1366000
002700
000460
00001
,
3926000
2733000
000067
006000
000440
,
6836000
485000
000450
001300
000072

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

׿
×
𝔽