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G = S32⋊Q8order 288 = 25·32

The semidirect product of S32 and Q8 acting via Q8/C4=C2

non-abelian, soluble, monomial

Aliases: S32⋊Q8, C4.13S3≀C2, (C3×C12).8D4, C3⋊Dic3.26D4, Dic3.D66C2, C321(C22⋊Q8), C6.D6.1C22, S32⋊C4.C2, (C4×S32).3C2, C2.4(C2×S3≀C2), (C3×C6).1(C2×D4), C3⋊S3.Q81C2, C3⋊S3.1(C2×Q8), C4⋊(C32⋊C4)⋊3C2, (C2×S32).5C22, C3⋊S3.1(C4○D4), (C2×C3⋊S3).1C23, (C4×C3⋊S3).27C22, (C2×C32⋊C4).4C22, SmallGroup(288,868)

Series: Derived Chief Lower central Upper central

C1C32C2×C3⋊S3 — S32⋊Q8
C1C32C3⋊S3C2×C3⋊S3C2×S32S32⋊C4 — S32⋊Q8
C32C2×C3⋊S3 — S32⋊Q8
C1C2C4

Generators and relations for S32⋊Q8
 G = < a,b,c,d,e,f | a3=b2=c3=d2=e4=1, f2=e2, bab=fcf-1=a-1, ac=ca, ad=da, ae=ea, faf-1=dcd=c-1, bc=cb, bd=db, be=eb, fbf-1=d, ce=ec, de=ed, fdf-1=b, fef-1=e-1 >

Subgroups: 600 in 120 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×4], C3 [×2], C4, C4 [×6], C22 [×5], S3 [×6], C6 [×4], C2×C4 [×8], Q8 [×2], C23, C32, Dic3 [×5], C12 [×5], D6 [×7], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×S3 [×2], C3⋊S3 [×2], C3×C6, Dic6 [×3], C4×S3 [×7], C2×Dic3, C2×C12, C3×Q8, C22×S3, C22⋊Q8, C3×Dic3 [×3], C3⋊Dic3, C3×C12, C32⋊C4 [×2], S32 [×2], S32, S3×C6, C2×C3⋊S3, S3×C2×C4, S3×Q8, S3×Dic3, C6.D6, C6.D6 [×2], C322Q8, C3×Dic6, S3×C12, C4×C3⋊S3, C2×C32⋊C4 [×2], C2×S32, S32⋊C4 [×2], C3⋊S3.Q8 [×2], C4⋊(C32⋊C4), Dic3.D6, C4×S32, S32⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, S3≀C2, C2×S3≀C2, S32⋊Q8

Character table of S32⋊Q8

 class 12A2B2C2D2E3A3B4A4B4C4D4E4F4G4H6A6B6C6D12A12B12C12D12E12F12G
 size 11669944266121218363644121244812122424
ρ1111111111111111111111111111    trivial
ρ211-1-111111-1-1111-1-111-1-1111-1-111    linear of order 2
ρ311111111-1-1-1-11-11-11111-1-1-1-1-11-1    linear of order 2
ρ411-1-11111-111-11-1-1111-1-1-1-1-1111-1    linear of order 2
ρ511111111-1-1-11-1-1-111111-1-1-1-1-1-11    linear of order 2
ρ611-1-11111-1111-1-11-111-1-1-1-1-111-11    linear of order 2
ρ711111111111-1-11-1-1111111111-1-1    linear of order 2
ρ811-1-111111-1-1-1-111111-1-1111-1-1-1-1    linear of order 2
ρ92200-2-22220000-20022002220000    orthogonal lifted from D4
ρ102200-2-222-200002002200-2-2-20000    orthogonal lifted from D4
ρ112-22-22-22200000000-2-2-220000000    symplectic lifted from Q8, Schur index 2
ρ122-2-222-22200000000-2-22-20000000    symplectic lifted from Q8, Schur index 2
ρ132-200-222202i-2i00000-2-2000002i-2i00    complex lifted from C4○D4
ρ142-200-22220-2i2i00000-2-200000-2i2i00    complex lifted from C4○D4
ρ1544-2-200-214-2-200000-211111-21100    orthogonal lifted from S3≀C2
ρ16442200-2142200000-21-1-111-2-1-100    orthogonal lifted from S3≀C2
ρ174400001-2-400-220001-20022-100-11    orthogonal lifted from C2×S3≀C2
ρ184400001-2400220001-200-2-2100-1-1    orthogonal lifted from S3≀C2
ρ194400001-2400-2-20001-200-2-210011    orthogonal lifted from S3≀C2
ρ20442200-21-4-2-200000-21-1-1-1-121100    orthogonal lifted from C2×S3≀C2
ρ2144-2-200-21-42200000-2111-1-12-1-100    orthogonal lifted from C2×S3≀C2
ρ224400001-2-4002-20001-20022-1001-1    orthogonal lifted from C2×S3≀C2
ρ234-4-2200-2102i-2i000002-1-11-3i3i0-ii00    complex faithful
ρ244-42-200-210-2i2i000002-11-1-3i3i0i-i00    complex faithful
ρ254-4-2200-210-2i2i000002-1-113i-3i0i-i00    complex faithful
ρ264-42-200-2102i-2i000002-11-13i-3i0-ii00    complex faithful
ρ278-800002-400000000-24000000000    symplectic faithful, Schur index 2

Permutation representations of S32⋊Q8
On 24 points - transitive group 24T641
Generators in S24
(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 20)(14 17)(15 18)(16 19)(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)

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

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

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

G:=TransitiveGroup(24,641);

On 24 points - transitive group 24T648
Generators in S24
(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 3)(2 4)(5 10)(6 11)(7 12)(8 9)(13 15)(14 16)(17 19)(18 20)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)
(5 7)(6 8)(9 11)(10 12)(13 18)(14 19)(15 20)(16 17)(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)

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

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

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

G:=TransitiveGroup(24,648);

Matrix representation of S32⋊Q8 in GL4(𝔽5) generated by

0020
0100
2040
0001
,
4030
0400
0010
0004
,
1000
0003
0010
0304
,
4000
0100
0040
0304
,
3000
0200
0030
0002
,
0003
0030
0300
3000
G:=sub<GL(4,GF(5))| [0,0,2,0,0,1,0,0,2,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,3,0,1,0,0,0,0,4],[1,0,0,0,0,0,0,3,0,0,1,0,0,3,0,4],[4,0,0,0,0,1,0,3,0,0,4,0,0,0,0,4],[3,0,0,0,0,2,0,0,0,0,3,0,0,0,0,2],[0,0,0,3,0,0,3,0,0,3,0,0,3,0,0,0] >;

S32⋊Q8 in GAP, Magma, Sage, TeX

S_3^2\rtimes Q_8
% in TeX

G:=Group("S3^2:Q8");
// GroupNames label

G:=SmallGroup(288,868);
// by ID

G=gap.SmallGroup(288,868);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,219,100,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^3=d^2=e^4=1,f^2=e^2,b*a*b=f*c*f^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=d*c*d=c^-1,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=d,c*e=e*c,d*e=e*d,f*d*f^-1=b,f*e*f^-1=e^-1>;
// generators/relations

Export

Character table of S32⋊Q8 in TeX

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