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G = S32⋊C4order 144 = 24·32

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

non-abelian, soluble, monomial

Aliases: S32⋊C4, C2.1S3≀C2, C3⋊S3.2D4, (C3×C6).1D4, C32⋊(C22⋊C4), C6.D64C2, (C2×S32).C2, C3⋊S3.2(C2×C4), (C2×C32⋊C4)⋊1C2, (C2×C3⋊S3).1C22, SmallGroup(144,115)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — S32⋊C4
C1C32C3⋊S3C2×C3⋊S3C2×S32 — S32⋊C4
C32C3⋊S3 — S32⋊C4
C1C2

Generators and relations for S32⋊C4
 G = < a,b,c,d,e | a3=b2=c3=d2=e4=1, bab=ece-1=a-1, ac=ca, ad=da, eae-1=dcd=c-1, bc=cb, bd=db, ebe-1=d, ede-1=b >

Subgroups: 278 in 58 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2 [×4], C3 [×2], C4 [×2], C22 [×5], S3 [×6], C6 [×4], C2×C4 [×2], C23, C32, Dic3, C12, D6 [×7], C2×C6, C22⋊C4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C4×S3, C22×S3, C3×Dic3, C32⋊C4, S32 [×2], S32, S3×C6, C2×C3⋊S3, C6.D6, C2×C32⋊C4, C2×S32, S32⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, S3≀C2, S32⋊C4

Character table of S32⋊C4

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D12A12B
 size 116699446618184412121212
ρ1111111111111111111    trivial
ρ211-1-1111111-1-111-1-111    linear of order 2
ρ311-1-11111-1-11111-1-1-1-1    linear of order 2
ρ411111111-1-1-1-11111-1-1    linear of order 2
ρ51-11-11-111-ii-ii-1-11-1-ii    linear of order 4
ρ61-1-111-111i-i-ii-1-1-11i-i    linear of order 4
ρ71-1-111-111-iii-i-1-1-11-ii    linear of order 4
ρ81-11-11-111i-ii-i-1-11-1i-i    linear of order 4
ρ92200-2-2220000220000    orthogonal lifted from D4
ρ102-200-22220000-2-20000    orthogonal lifted from D4
ρ114-42-2001-200002-1-1100    orthogonal faithful
ρ124-4-22001-200002-11-100    orthogonal faithful
ρ13440000-21-2-2001-20011    orthogonal lifted from S3≀C2
ρ144422001-20000-21-1-100    orthogonal lifted from S3≀C2
ρ15440000-2122001-200-1-1    orthogonal lifted from S3≀C2
ρ1644-2-2001-20000-211100    orthogonal lifted from S3≀C2
ρ174-40000-212i-2i00-1200-ii    complex faithful
ρ184-40000-21-2i2i00-1200i-i    complex faithful

Permutation representations of S32⋊C4
On 12 points - transitive group 12T79
Generators in S12
(2 5 10)(4 7 12)
(5 10)(7 12)
(1 9 8)(3 11 6)
(6 11)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)

G:=sub<Sym(12)| (2,5,10)(4,7,12), (5,10)(7,12), (1,9,8)(3,11,6), (6,11)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;

G:=Group( (2,5,10)(4,7,12), (5,10)(7,12), (1,9,8)(3,11,6), (6,11)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12) );

G=PermutationGroup([(2,5,10),(4,7,12)], [(5,10),(7,12)], [(1,9,8),(3,11,6)], [(6,11),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)])

G:=TransitiveGroup(12,79);

On 12 points - transitive group 12T80
Generators in S12
(1 8 9)(3 6 11)
(2 4)(5 7)(6 11)(8 9)(10 12)
(2 10 5)(4 12 7)
(1 3)(5 10)(6 8)(7 12)(9 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)

G:=sub<Sym(12)| (1,8,9)(3,6,11), (2,4)(5,7)(6,11)(8,9)(10,12), (2,10,5)(4,12,7), (1,3)(5,10)(6,8)(7,12)(9,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;

G:=Group( (1,8,9)(3,6,11), (2,4)(5,7)(6,11)(8,9)(10,12), (2,10,5)(4,12,7), (1,3)(5,10)(6,8)(7,12)(9,11), (1,2,3,4)(5,6,7,8)(9,10,11,12) );

G=PermutationGroup([(1,8,9),(3,6,11)], [(2,4),(5,7),(6,11),(8,9),(10,12)], [(2,10,5),(4,12,7)], [(1,3),(5,10),(6,8),(7,12),(9,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)])

G:=TransitiveGroup(12,80);

On 24 points - transitive group 24T213
Generators in S24
(1 15 20)(2 16 17)(3 13 18)(4 14 19)(5 23 10)(6 11 24)(7 21 12)(8 9 22)
(1 22)(2 23)(3 24)(4 21)(5 16)(6 18)(7 14)(8 20)(9 15)(10 17)(11 13)(12 19)
(1 20 15)(2 17 16)(3 18 13)(4 19 14)(5 23 10)(6 11 24)(7 21 12)(8 9 22)
(1 22)(2 23)(3 24)(4 21)(5 17)(6 13)(7 19)(8 15)(9 20)(10 16)(11 18)(12 14)
(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)| (1,15,20)(2,16,17)(3,13,18)(4,14,19)(5,23,10)(6,11,24)(7,21,12)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,16)(6,18)(7,14)(8,20)(9,15)(10,17)(11,13)(12,19), (1,20,15)(2,17,16)(3,18,13)(4,19,14)(5,23,10)(6,11,24)(7,21,12)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,17)(6,13)(7,19)(8,15)(9,20)(10,16)(11,18)(12,14), (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( (1,15,20)(2,16,17)(3,13,18)(4,14,19)(5,23,10)(6,11,24)(7,21,12)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,16)(6,18)(7,14)(8,20)(9,15)(10,17)(11,13)(12,19), (1,20,15)(2,17,16)(3,18,13)(4,19,14)(5,23,10)(6,11,24)(7,21,12)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,17)(6,13)(7,19)(8,15)(9,20)(10,16)(11,18)(12,14), (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([(1,15,20),(2,16,17),(3,13,18),(4,14,19),(5,23,10),(6,11,24),(7,21,12),(8,9,22)], [(1,22),(2,23),(3,24),(4,21),(5,16),(6,18),(7,14),(8,20),(9,15),(10,17),(11,13),(12,19)], [(1,20,15),(2,17,16),(3,18,13),(4,19,14),(5,23,10),(6,11,24),(7,21,12),(8,9,22)], [(1,22),(2,23),(3,24),(4,21),(5,17),(6,13),(7,19),(8,15),(9,20),(10,16),(11,18),(12,14)], [(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(24,213);

On 24 points - transitive group 24T215
Generators in S24
(1 18 13)(2 19 14)(3 20 15)(4 17 16)(5 21 10)(6 11 22)(7 23 12)(8 9 24)
(1 22)(2 21)(3 24)(4 23)(5 19)(6 13)(7 17)(8 15)(9 20)(10 14)(11 18)(12 16)
(1 13 18)(2 14 19)(3 15 20)(4 16 17)(5 21 10)(6 11 22)(7 23 12)(8 9 24)
(1 24)(2 23)(3 22)(4 21)(5 16)(6 20)(7 14)(8 18)(9 13)(10 17)(11 15)(12 19)
(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)| (1,18,13)(2,19,14)(3,20,15)(4,17,16)(5,21,10)(6,11,22)(7,23,12)(8,9,24), (1,22)(2,21)(3,24)(4,23)(5,19)(6,13)(7,17)(8,15)(9,20)(10,14)(11,18)(12,16), (1,13,18)(2,14,19)(3,15,20)(4,16,17)(5,21,10)(6,11,22)(7,23,12)(8,9,24), (1,24)(2,23)(3,22)(4,21)(5,16)(6,20)(7,14)(8,18)(9,13)(10,17)(11,15)(12,19), (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( (1,18,13)(2,19,14)(3,20,15)(4,17,16)(5,21,10)(6,11,22)(7,23,12)(8,9,24), (1,22)(2,21)(3,24)(4,23)(5,19)(6,13)(7,17)(8,15)(9,20)(10,14)(11,18)(12,16), (1,13,18)(2,14,19)(3,15,20)(4,16,17)(5,21,10)(6,11,22)(7,23,12)(8,9,24), (1,24)(2,23)(3,22)(4,21)(5,16)(6,20)(7,14)(8,18)(9,13)(10,17)(11,15)(12,19), (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([(1,18,13),(2,19,14),(3,20,15),(4,17,16),(5,21,10),(6,11,22),(7,23,12),(8,9,24)], [(1,22),(2,21),(3,24),(4,23),(5,19),(6,13),(7,17),(8,15),(9,20),(10,14),(11,18),(12,16)], [(1,13,18),(2,14,19),(3,15,20),(4,16,17),(5,21,10),(6,11,22),(7,23,12),(8,9,24)], [(1,24),(2,23),(3,22),(4,21),(5,16),(6,20),(7,14),(8,18),(9,13),(10,17),(11,15),(12,19)], [(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(24,215);

On 24 points - transitive group 24T265
Generators in S24
(2 10 6)(4 12 8)(14 24 20)(16 22 18)
(1 3)(2 20)(4 18)(5 7)(6 14)(8 16)(9 11)(10 24)(12 22)(13 15)(17 19)(21 23)
(1 5 9)(3 7 11)(13 19 23)(15 17 21)
(1 19)(2 4)(3 17)(5 13)(6 8)(7 15)(9 23)(10 12)(11 21)(14 16)(18 20)(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)

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

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

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

G:=TransitiveGroup(24,265);

On 24 points - transitive group 24T266
Generators in S24
(1 9 5)(3 11 7)(13 23 19)(15 21 17)
(2 18)(4 20)(5 9)(6 22)(7 11)(8 24)(10 16)(12 14)(13 23)(15 21)
(2 6 10)(4 8 12)(14 20 24)(16 18 22)
(1 17)(3 19)(5 21)(6 10)(7 23)(8 12)(9 15)(11 13)(14 24)(16 22)
(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)| (1,9,5)(3,11,7)(13,23,19)(15,21,17), (2,18)(4,20)(5,9)(6,22)(7,11)(8,24)(10,16)(12,14)(13,23)(15,21), (2,6,10)(4,8,12)(14,20,24)(16,18,22), (1,17)(3,19)(5,21)(6,10)(7,23)(8,12)(9,15)(11,13)(14,24)(16,22), (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( (1,9,5)(3,11,7)(13,23,19)(15,21,17), (2,18)(4,20)(5,9)(6,22)(7,11)(8,24)(10,16)(12,14)(13,23)(15,21), (2,6,10)(4,8,12)(14,20,24)(16,18,22), (1,17)(3,19)(5,21)(6,10)(7,23)(8,12)(9,15)(11,13)(14,24)(16,22), (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([(1,9,5),(3,11,7),(13,23,19),(15,21,17)], [(2,18),(4,20),(5,9),(6,22),(7,11),(8,24),(10,16),(12,14),(13,23),(15,21)], [(2,6,10),(4,8,12),(14,20,24),(16,18,22)], [(1,17),(3,19),(5,21),(6,10),(7,23),(8,12),(9,15),(11,13),(14,24),(16,22)], [(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(24,266);

On 24 points - transitive group 24T267
Generators in S24
(2 16 17)(4 14 19)(5 23 10)(7 21 12)
(1 22)(2 23)(3 24)(4 21)(5 16)(6 18)(7 14)(8 20)(9 15)(10 17)(11 13)(12 19)
(1 20 15)(3 18 13)(6 11 24)(8 9 22)
(1 22)(2 23)(3 24)(4 21)(5 17)(6 13)(7 19)(8 15)(9 20)(10 16)(11 18)(12 14)
(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)| (2,16,17)(4,14,19)(5,23,10)(7,21,12), (1,22)(2,23)(3,24)(4,21)(5,16)(6,18)(7,14)(8,20)(9,15)(10,17)(11,13)(12,19), (1,20,15)(3,18,13)(6,11,24)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,17)(6,13)(7,19)(8,15)(9,20)(10,16)(11,18)(12,14), (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( (2,16,17)(4,14,19)(5,23,10)(7,21,12), (1,22)(2,23)(3,24)(4,21)(5,16)(6,18)(7,14)(8,20)(9,15)(10,17)(11,13)(12,19), (1,20,15)(3,18,13)(6,11,24)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,17)(6,13)(7,19)(8,15)(9,20)(10,16)(11,18)(12,14), (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([(2,16,17),(4,14,19),(5,23,10),(7,21,12)], [(1,22),(2,23),(3,24),(4,21),(5,16),(6,18),(7,14),(8,20),(9,15),(10,17),(11,13),(12,19)], [(1,20,15),(3,18,13),(6,11,24),(8,9,22)], [(1,22),(2,23),(3,24),(4,21),(5,17),(6,13),(7,19),(8,15),(9,20),(10,16),(11,18),(12,14)], [(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(24,267);

On 24 points - transitive group 24T268
Generators in S24
(2 19 14)(4 17 16)(5 21 10)(7 23 12)
(1 22)(2 21)(3 24)(4 23)(5 19)(6 13)(7 17)(8 15)(9 20)(10 14)(11 18)(12 16)
(1 13 18)(3 15 20)(6 11 22)(8 9 24)
(1 24)(2 23)(3 22)(4 21)(5 16)(6 20)(7 14)(8 18)(9 13)(10 17)(11 15)(12 19)
(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)| (2,19,14)(4,17,16)(5,21,10)(7,23,12), (1,22)(2,21)(3,24)(4,23)(5,19)(6,13)(7,17)(8,15)(9,20)(10,14)(11,18)(12,16), (1,13,18)(3,15,20)(6,11,22)(8,9,24), (1,24)(2,23)(3,22)(4,21)(5,16)(6,20)(7,14)(8,18)(9,13)(10,17)(11,15)(12,19), (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( (2,19,14)(4,17,16)(5,21,10)(7,23,12), (1,22)(2,21)(3,24)(4,23)(5,19)(6,13)(7,17)(8,15)(9,20)(10,14)(11,18)(12,16), (1,13,18)(3,15,20)(6,11,22)(8,9,24), (1,24)(2,23)(3,22)(4,21)(5,16)(6,20)(7,14)(8,18)(9,13)(10,17)(11,15)(12,19), (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([(2,19,14),(4,17,16),(5,21,10),(7,23,12)], [(1,22),(2,21),(3,24),(4,23),(5,19),(6,13),(7,17),(8,15),(9,20),(10,14),(11,18),(12,16)], [(1,13,18),(3,15,20),(6,11,22),(8,9,24)], [(1,24),(2,23),(3,22),(4,21),(5,16),(6,20),(7,14),(8,18),(9,13),(10,17),(11,15),(12,19)], [(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(24,268);

S32⋊C4 is a maximal subgroup of
S32⋊Q8  C4.4S3≀C2  C4×S3≀C2  S32⋊D4  C62.9D4  D6≀C2  C62⋊D4  C3⋊S3.2D12  S32⋊Dic3  (C3×C6).8D12
S32⋊C4 is a maximal quotient of
S32⋊C8  C4.S3≀C2  (C3×C12).D4  C3⋊S3.2D8  C3⋊S3.2Q16  C32⋊C4≀C2  C62.D4  C62.2D4  C62.3D4  C62.4D4  Dic3≀C2  C32⋊D6⋊C4  C3⋊S3.2D12  S32⋊Dic3  (C3×C6).8D12

Polynomial with Galois group S32⋊C4 over ℚ
actionf(x)Disc(f)
12T79x12-16x10+96x8-260x6+288x4-64x2+2235·1032·1912
12T80x12-16x9-36x8-24x7-136x6+210x4-128x3-72x2+96x-8-251·324·5692·2578672

Matrix representation of S32⋊C4 in GL4(ℤ) generated by

1000
0100
00-1-1
0010
,
-1000
0-100
00-10
0011
,
0100
-1-100
0010
0001
,
-1000
1100
00-10
000-1
,
0010
0001
-1000
0-100
G:=sub<GL(4,Integers())| [1,0,0,0,0,1,0,0,0,0,-1,1,0,0,-1,0],[-1,0,0,0,0,-1,0,0,0,0,-1,1,0,0,0,1],[0,-1,0,0,1,-1,0,0,0,0,1,0,0,0,0,1],[-1,1,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[0,0,-1,0,0,0,0,-1,1,0,0,0,0,1,0,0] >;

S32⋊C4 in GAP, Magma, Sage, TeX

S_3^2\rtimes C_4
% in TeX

G:=Group("S3^2:C4");
// GroupNames label

G:=SmallGroup(144,115);
// by ID

G=gap.SmallGroup(144,115);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,73,55,964,730,256,299,881]);
// Polycyclic

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

Export

Character table of S32⋊C4 in TeX

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