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G = C23.7S4order 192 = 26·3

1st non-split extension by C23 of S4 acting via S4/C22=S3

non-abelian, soluble

Aliases: C23.7S4, C22.CSU2(𝔽3), C2.2(C42⋊S3), C2.C42.S3, C23.3A4.2C2, SmallGroup(192,180)

Series: Derived Chief Lower central Upper central

C1C2C2.C42C23.3A4 — C23.7S4
C1C2C23C2.C42C23.3A4 — C23.7S4
C23.3A4 — C23.7S4
C1C2

Generators and relations for C23.7S4
 G = < a,b,c,d,e,f,g | a2=b2=c2=f3=1, d2=gag-1=fbf-1=abc, e2=faf-1=b, g2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, geg-1=be=eb, bg=gb, ede-1=cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=cde, gdg-1=de, fef-1=bcd, gfg-1=f-1 >

3C2
3C2
16C3
3C22
3C22
6C4
6C4
12C4
12C4
16C6
3C2×C4
3C2×C4
6C2×C4
6C2×C4
12C2×C4
12Q8
12C8
4A4
16Dic3
3C22×C4
3C4⋊C4
3C2×Q8
6C22⋊C4
6C2×C8
6C4⋊C4
4C2×A4
3C22⋊Q8
3C22⋊C8
4A4⋊C4
3C23.31D4

Character table of C23.7S4

 class 12A2B2C34A4B4C4D4E68A8B8C8D
 size 113332661224243212121212
ρ1111111111111111    trivial
ρ211111111-1-11-1-1-1-1    linear of order 2
ρ32222-122200-10000    orthogonal lifted from S3
ρ42-2-22-1000001-222-2    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ52-2-22-10000012-2-22    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ633330-1-1-1-1-101111    orthogonal lifted from S4
ρ733330-1-1-1110-1-1-1-1    orthogonal lifted from S4
ρ833-1-10-1+2i-1-2i11-10ii-i-i    complex lifted from C42⋊S3
ρ933-1-10-1-2i-1+2i1-110ii-i-i    complex lifted from C42⋊S3
ρ1033-1-10-1+2i-1-2i1-110-i-iii    complex lifted from C42⋊S3
ρ1133-1-10-1-2i-1+2i11-10-i-iii    complex lifted from C42⋊S3
ρ124-4-44100000-10000    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ1366-2-2022-20000000    orthogonal lifted from C42⋊S3
ρ146-62-20000000-2--2-2--2    complex faithful
ρ156-62-20000000--2-2--2-2    complex faithful

Permutation representations of C23.7S4
On 24 points - transitive group 24T316
Generators in S24
(3 4)(5 6)(7 10)(8 9)
(1 2)(11 15)(12 16)(13 14)
(1 2)(3 4)(5 6)(7 10)(8 9)(11 15)(12 16)(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 14 2 13)(3 5)(4 6)(7 10)(11 12 15 16)(17 19)(21 22)(23 24)
(1 9 24)(2 8 22)(3 17 15)(4 19 11)(5 20 16)(6 18 12)(7 23 14)(10 21 13)
(1 16 2 12)(3 21 4 23)(5 22 6 24)(7 17 10 19)(8 18 9 20)(11 14 15 13)

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

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

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

G:=TransitiveGroup(24,316);

On 24 points - transitive group 24T427
Generators in S24
(1 4)(2 3)(5 6)(7 8)(13 24)(14 21)(15 22)(16 23)
(1 2)(3 4)(5 8)(6 7)(9 18)(10 19)(11 20)(12 17)
(1 3)(2 4)(5 7)(6 8)(9 20)(10 17)(11 18)(12 19)(13 22)(14 23)(15 24)(16 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 5 2 8)(3 7 4 6)(9 17 18 12)(10 11 19 20)(13 22)(15 24)
(1 15 19)(2 22 17)(3 24 12)(4 13 10)(5 14 18)(6 16 9)(7 23 11)(8 21 20)
(1 17 3 10)(2 12 4 19)(5 9 7 20)(6 11 8 18)(13 15 22 24)(14 16 23 21)

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

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

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

G:=TransitiveGroup(24,427);

On 24 points - transitive group 24T429
Generators in S24
(1 3)(2 4)(5 7)(6 8)(17 23)(18 24)(19 21)(20 22)
(1 2)(3 4)(5 8)(6 7)(9 16)(10 13)(11 14)(12 15)
(1 4)(2 3)(5 6)(7 8)(9 14)(10 15)(11 16)(12 13)(17 21)(18 22)(19 23)(20 24)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8 2 5)(3 6 4 7)(9 12 16 15)(10 14 13 11)(17 23)(18 20)(19 21)(22 24)
(1 24 14)(2 18 16)(3 22 11)(4 20 9)(5 23 15)(6 19 10)(7 21 12)(8 17 13)
(1 10 4 15)(2 13 3 12)(5 14 6 9)(7 16 8 11)(17 22 21 18)(19 20 23 24)

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

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

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

G:=TransitiveGroup(24,429);

Matrix representation of C23.7S4 in GL5(𝔽73)

720000
072000
007200
000720
00001
,
720000
072000
00100
000720
000072
,
720000
072000
00100
00010
00001
,
6617000
407000
004600
000720
000046
,
2735000
046000
00100
000270
000046
,
1134000
2461000
000460
00001
002700
,
652000
48000
00100
00001
00010

G:=sub<GL(5,GF(73))| [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],[72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[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],[66,40,0,0,0,17,7,0,0,0,0,0,46,0,0,0,0,0,72,0,0,0,0,0,46],[27,0,0,0,0,35,46,0,0,0,0,0,1,0,0,0,0,0,27,0,0,0,0,0,46],[11,24,0,0,0,34,61,0,0,0,0,0,0,0,27,0,0,46,0,0,0,0,0,1,0],[65,4,0,0,0,2,8,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C23.7S4 in GAP, Magma, Sage, TeX

C_2^3._7S_4
% in TeX

G:=Group("C2^3.7S4");
// GroupNames label

G:=SmallGroup(192,180);
// by ID

G=gap.SmallGroup(192,180);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,57,254,1143,268,171,934,521,80,2524,2531,3540]);
// Polycyclic

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

Export

Subgroup lattice of C23.7S4 in TeX
Character table of C23.7S4 in TeX

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