S3^2:C4 - GroupNames

G = S₃²⋊C₄ order 144 = 2⁴·3²

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

Aliases: S₃²⋊C₄, C₂.₁S₃≀C₂, C₃⋊S₃.₂D₄, (C₃×C₆).₁D₄, C₃²⋊(C₂²⋊C₄), C₆.D₆⋊₄C₂, (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

Derived series

C₁ — C₃² — C₃⋊S₃ — S₃²⋊C₄

Chief series

C₁ — C₃² — C₃⋊S₃ — C₂×C₃⋊S₃ — C₂×S₃² — S₃²⋊C₄

Lower central

C₃² — C₃⋊S₃ — S₃²⋊C₄

Upper central

C₁ — C₂

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, C₃, C₄, C₂², S₃, C₆, C₂×C₄, C₂³, C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₂²⋊C₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₄×S₃, C₂²×S₃, C₃×Dic₃, C₃²⋊C₄, S₃², S₃², S₃×C₆, C₂×C₃⋊S₃, C₆.D₆, C₂×C₃²⋊C₄, C₂×S₃², S₃²⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, S₃≀C₂, S₃²⋊C₄

Character table of S₃²⋊C₄

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	4D	6A	6B	6C	6D	12A	12B
size	1	1	6	6	9	9	4	4	6	6	18	18	4	4	12	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	1	1	linear of order 2
ρ₃	1	1	-1	-1	1	1	1	1	-1	-1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	-1	-1	linear of order 2
ρ₅	1	-1	1	-1	1	-1	1	1	-i	i	-i	i	-1	-1	1	-1	-i	i	linear of order 4
ρ₆	1	-1	-1	1	1	-1	1	1	i	-i	-i	i	-1	-1	-1	1	i	-i	linear of order 4
ρ₇	1	-1	-1	1	1	-1	1	1	-i	i	i	-i	-1	-1	-1	1	-i	i	linear of order 4
ρ₈	1	-1	1	-1	1	-1	1	1	i	-i	i	-i	-1	-1	1	-1	i	-i	linear of order 4
ρ₉	2	2	0	0	-2	-2	2	2	0	0	0	0	2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	0	0	-2	2	2	2	0	0	0	0	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	4	-4	2	-2	0	0	1	-2	0	0	0	0	2	-1	-1	1	0	0	orthogonal faithful
ρ₁₂	4	-4	-2	2	0	0	1	-2	0	0	0	0	2	-1	1	-1	0	0	orthogonal faithful
ρ₁₃	4	4	0	0	0	0	-2	1	-2	-2	0	0	1	-2	0	0	1	1	orthogonal lifted from S₃≀C₂
ρ₁₄	4	4	2	2	0	0	1	-2	0	0	0	0	-2	1	-1	-1	0	0	orthogonal lifted from S₃≀C₂
ρ₁₅	4	4	0	0	0	0	-2	1	2	2	0	0	1	-2	0	0	-1	-1	orthogonal lifted from S₃≀C₂
ρ₁₆	4	4	-2	-2	0	0	1	-2	0	0	0	0	-2	1	1	1	0	0	orthogonal lifted from S₃≀C₂
ρ₁₇	4	-4	0	0	0	0	-2	1	2i	-2i	0	0	-1	2	0	0	-i	i	complex faithful
ρ₁₈	4	-4	0	0	0	0	-2	1	-2i	2i	0	0	-1	2	0	0	i	-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

action	f(x)	Disc(f)
12T79	x¹²-16x¹⁰+96x⁸-260x⁶+288x⁴-64x²+2	2³⁵·103²·191²
12T80	x¹²-16x⁹-36x⁸-24x⁷-136x⁶+210x⁴-128x³-72x²+96x-8	-2⁵¹·3²⁴·569²·257867²

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

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	0
0	0	1	1

0	1	0	0
-1	-1	0	0
0	0	1	0
0	0	0	1

-1	0	0	0
1	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

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

G = S32⋊C4 order 144 = 24·32

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

G = S₃²⋊C₄ order 144 = 2⁴·3²

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