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G = C25.S3order 192 = 26·3

1st non-split extension by C25 of S3 acting faithfully

non-abelian, soluble, monomial

Aliases: C25.1S3, C24.6D6, C23.17S4, C242Dic3, (C22×A4)⋊2C4, (C2×A4).11D4, A42(C22⋊C4), C222(A4⋊C4), (C23×A4).2C2, C22.21(C2×S4), C2.3(A4⋊D4), C22⋊(C6.D4), C23.5(C2×Dic3), C23.21(C3⋊D4), (C22×A4).7C22, (C2×A4⋊C4)⋊2C2, C2.11(C2×A4⋊C4), (C2×A4).9(C2×C4), SmallGroup(192,991)

Series: Derived Chief Lower central Upper central

C1C22C2×A4 — C25.S3
C1C22A4C2×A4C22×A4C2×A4⋊C4 — C25.S3
A4C2×A4 — C25.S3
C1C22C23

Generators and relations for C25.S3
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=1, g2=b, ab=ba, gag-1=ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, fdf-1=gdg-1=e, geg-1=d, gfg-1=f-1 >

Subgroups: 714 in 173 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C22 [×2], C22 [×2], C22 [×34], C6 [×5], C2×C4 [×12], C23 [×2], C23 [×2], C23 [×34], Dic3 [×2], A4, C2×C6 [×5], C22⋊C4 [×12], C22×C4 [×4], C24, C24 [×2], C24 [×8], C2×Dic3 [×2], C2×A4, C2×A4 [×2], C2×A4 [×2], C22×C6, C2×C22⋊C4 [×6], C25, C6.D4, A4⋊C4 [×2], C22×A4, C22×A4 [×2], C22×A4 [×2], C243C4, C2×A4⋊C4 [×2], C23×A4, C25.S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], S4, C6.D4, A4⋊C4 [×2], C2×S4, C2×A4⋊C4, A4⋊D4 [×2], C25.S3

Character table of C25.S3

 class 12A2B2C2D2E2F2G2H2I2J2K34A4B4C4D4E4F4G4H6A6B6C6D6E6F6G
 size 111122333366812121212121212128888888
ρ11111111111111111111111111111    trivial
ρ21111-1-11111-1-11-1111-1-1-11-1-111-1-11    linear of order 2
ρ31111111111111-1-1-1-1-1-1-1-11111111    linear of order 2
ρ41111-1-11111-1-111-1-1-1111-1-1-111-1-11    linear of order 2
ρ511-1-11-1-1-111-111ii-i-i-i-iii-1-1-1111-1    linear of order 4
ρ611-1-11-1-1-111-111-i-iiiii-i-i-1-1-1111-1    linear of order 4
ρ711-1-1-11-1-1111-11i-iii-i-ii-i11-11-1-1-1    linear of order 4
ρ811-1-1-11-1-1111-11-ii-i-iii-ii11-11-1-1-1    linear of order 4
ρ92222-2-22222-2-2-10000000011-1-111-1    orthogonal lifted from D6
ρ102-2-2200-22-2200200000000002-200-2    orthogonal lifted from D4
ρ11222222222222-100000000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ122-22-2002-2-220020000000000-2-2002    orthogonal lifted from D4
ρ1322-2-2-22-2-2222-2-100000000-1-11-1111    symplectic lifted from Dic3, Schur index 2
ρ1422-2-22-2-2-222-22-100000000111-1-1-11    symplectic lifted from Dic3, Schur index 2
ρ152-2-2200-22-2200-100000000-3--3-11--3-31    complex lifted from C3⋊D4
ρ162-2-2200-22-2200-100000000--3-3-11-3--31    complex lifted from C3⋊D4
ρ172-22-2002-2-2200-100000000-3--311-3--3-1    complex lifted from C3⋊D4
ρ182-22-2002-2-2200-100000000--3-311--3-3-1    complex lifted from C3⋊D4
ρ19333333-1-1-1-1-1-10-1-11-11-1110000000    orthogonal lifted from S4
ρ203333-3-3-1-1-1-1110-11-111-11-10000000    orthogonal lifted from C2×S4
ρ21333333-1-1-1-1-1-1011-11-11-1-10000000    orthogonal lifted from S4
ρ223333-3-3-1-1-1-11101-11-1-11-110000000    orthogonal lifted from C2×S4
ρ2333-3-33-311-1-11-10iii-ii-i-i-i0000000    complex lifted from A4⋊C4
ρ2433-3-33-311-1-11-10-i-i-ii-iiii0000000    complex lifted from A4⋊C4
ρ2533-3-3-3311-1-1-110-iii-i-iii-i0000000    complex lifted from A4⋊C4
ρ2633-3-3-3311-1-1-110i-i-iii-i-ii0000000    complex lifted from A4⋊C4
ρ276-6-66002-22-2000000000000000000    orthogonal lifted from A4⋊D4
ρ286-66-600-222-2000000000000000000    orthogonal lifted from A4⋊D4

Permutation representations of C25.S3
On 24 points - transitive group 24T402
Generators in S24
(2 12)(4 10)(5 16)(7 14)(18 22)(20 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 4)(3 9)(5 16)(6 15)(7 14)(8 13)(10 12)(17 19)(18 24)(20 22)(21 23)
(1 3)(2 12)(4 10)(5 14)(6 13)(7 16)(8 15)(9 11)(17 23)(18 20)(19 21)(22 24)
(1 19 6)(2 7 20)(3 17 8)(4 5 18)(9 21 15)(10 16 22)(11 23 13)(12 14 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)

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

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

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

G:=TransitiveGroup(24,402);

On 24 points - transitive group 24T403
Generators in S24
(1 3)(2 5)(4 7)(6 8)(9 11)(10 13)(12 15)(14 16)(17 24)(18 20)(19 22)(21 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 6)(2 7)(3 8)(4 5)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(2 7)(4 5)(9 14)(10 15)(11 16)(12 13)(18 23)(20 21)
(1 6)(3 8)(9 14)(10 15)(11 16)(12 13)(17 22)(19 24)
(1 9 23)(2 24 10)(3 11 21)(4 22 12)(5 17 13)(6 14 18)(7 19 15)(8 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)

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

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

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

G:=TransitiveGroup(24,403);

On 24 points - transitive group 24T404
Generators in S24
(1 23)(2 4)(3 21)(5 17)(6 8)(7 19)(9 16)(10 12)(11 14)(13 15)(18 20)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 17)(8 18)(9 14)(10 15)(11 16)(12 13)
(2 24)(4 22)(5 17)(7 19)(9 16)(10 13)(11 14)(12 15)
(1 23)(3 21)(6 18)(8 20)(9 16)(10 13)(11 14)(12 15)
(1 11 19)(2 20 12)(3 9 17)(4 18 10)(5 21 16)(6 13 22)(7 23 14)(8 15 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)

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

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

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

G:=TransitiveGroup(24,404);

Matrix representation of C25.S3 in GL7(𝔽13)

12000000
01200000
0010000
00212000
0000100
0000010
0000001
,
12000000
01200000
00120000
00012000
0000100
0000010
0000001
,
1000000
0100000
00120000
00012000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
00001200
00000120
0000001
,
1000000
0100000
0010000
0001000
0000100
00000120
00000012
,
121200000
1000000
0010000
0001000
00000120
00000012
0000100
,
0500000
5000000
0058000
0008000
0000001
0000010
0000100

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0],[0,5,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;

C25.S3 in GAP, Magma, Sage, TeX

C_2^5.S_3
% in TeX

G:=Group("C2^5.S3");
// GroupNames label

G:=SmallGroup(192,991);
// by ID

G=gap.SmallGroup(192,991);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,28,141,1124,4037,285,2358,475]);
// Polycyclic

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

Export

Character table of C25.S3 in TeX

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