C4^2:S3 - GroupNames

class	1	2A	2B	3	4A	4B	4C	4D	8A	8B
size	1	3	12	32	3	3	6	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	2	2	0	-1	2	2	2	0	0	0	orthogonal lifted from S₃
ρ₄	3	3	-1	0	-1	-1	-1	-1	1	1	orthogonal lifted from S₄
ρ₅	3	3	1	0	-1	-1	-1	1	-1	-1	orthogonal lifted from S₄
ρ₆	3	-1	1	0	-1+2i	-1-2i	1	-1	-i	i	complex faithful
ρ₇	3	-1	-1	0	-1+2i	-1-2i	1	1	i	-i	complex faithful
ρ₈	3	-1	-1	0	-1-2i	-1+2i	1	1	-i	i	complex faithful
ρ₉	3	-1	1	0	-1-2i	-1+2i	1	-1	i	-i	complex faithful
ρ₁₀	6	-2	0	0	2	2	-2	0	0	0	orthogonal faithful

Permutation representations of C₄²⋊S₃
►On 12 points - transitive group 12T62

Generators in S₁₂

(1 2)(3 4)(5 6 7 8)(9 10 11 12)

(1 4 2 3)(5 7)(6 8)(9 10 11 12)

(1 11 7)(2 9 5)(3 10 6)(4 12 8)

(1 7)(2 5)(3 6)(4 8)

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

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

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

G:=TransitiveGroup(12,62);

►On 12 points - transitive group 12T63

Generators in S₁₂

(1 2)(3 4)(5 6 7 8)(9 10 11 12)

(1 4 2 3)(5 7)(6 8)(9 10 11 12)

(1 11 7)(2 9 5)(3 10 6)(4 12 8)

(1 5)(2 7)(3 8)(4 6)(9 11)(10 12)

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

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

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

G:=TransitiveGroup(12,63);

►On 12 points - transitive group 12T64

Generators in S₁₂

(5 6 7 8)(9 10 11 12)

(1 3 4 2)(9 12 11 10)

(1 10 7)(2 9 8)(3 11 6)(4 12 5)

(1 7)(2 6)(3 8)(4 5)(9 11)

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

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

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

G:=TransitiveGroup(12,64);

►On 12 points - transitive group 12T65

Generators in S₁₂

(5 6 7 8)(9 10 11 12)

(1 3 4 2)(9 12 11 10)

(1 10 5)(2 9 6)(3 11 8)(4 12 7)

(1 6)(2 5)(3 7)(4 8)(9 10)(11 12)

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

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

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

G:=TransitiveGroup(12,65);

►On 16 points - transitive group 16T195

Generators in S₁₆

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 10 13 8)(2 11 14 5)(3 12 15 6)(4 9 16 7)

(2 7 10)(3 15 13)(4 11 8)(5 6 14)(9 12 16)

(2 10)(3 13)(4 8)(5 9)(6 16)(12 14)

G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10,13,8)(2,11,14,5)(3,12,15,6)(4,9,16,7), (2,7,10)(3,15,13)(4,11,8)(5,6,14)(9,12,16), (2,10)(3,13)(4,8)(5,9)(6,16)(12,14)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10,13,8)(2,11,14,5)(3,12,15,6)(4,9,16,7), (2,7,10)(3,15,13)(4,11,8)(5,6,14)(9,12,16), (2,10)(3,13)(4,8)(5,9)(6,16)(12,14) );

G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,10,13,8),(2,11,14,5),(3,12,15,6),(4,9,16,7)], [(2,7,10),(3,15,13),(4,11,8),(5,6,14),(9,12,16)], [(2,10),(3,13),(4,8),(5,9),(6,16),(12,14)]])

G:=TransitiveGroup(16,195);

►On 24 points - transitive group 24T191

Generators in S₂₄

(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 3 2 4)(5 8 6 7)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 23)(22 24)

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

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

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

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

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

G:=TransitiveGroup(24,191);

►On 24 points - transitive group 24T192

Generators in S₂₄

(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 6 8 3)(2 5 7 4)(9 24 11 22)(10 21 12 23)(13 19)(14 20)(15 17)(16 18)

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

(1 17)(2 15)(3 20)(4 14)(5 16)(6 18)(7 13)(8 19)(10 24)(12 22)

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

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

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

G:=TransitiveGroup(24,192);

►On 24 points - transitive group 24T193

Generators in S₂₄

(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 3 6 7)(2 4 5 8)(9 15 11 13)(10 16 12 14)(17 21)(18 22)(19 23)(20 24)

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

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

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

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

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

G:=TransitiveGroup(24,193);

►On 24 points - transitive group 24T194

Generators in S₂₄

(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 3 2 4)(5 7 8 6)(13 16 15 14)(17 20 19 18)

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

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

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

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

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

G:=TransitiveGroup(24,194);

C₄²⋊S₃ is a maximal subgroup of C₄²⋊D₆ (C₄×C₁₂)⋊S₃ C₄²⋊D₁₅
C₄²⋊S₃ is a maximal quotient of C₂³.₇S₄ C₂³.₈S₄ C₂³.₉S₄ C₄²⋊D₉ (C₄×C₁₂)⋊S₃ C₄²⋊D₁₅

Polynomial with Galois group C₄²⋊S₃ over ℚ

action	f(x)	Disc(f)
12T62	x¹²-13x¹⁰+59x⁸-109x⁶+73x⁴-16x²+1	2²⁰·19⁴·103⁴
12T63	x¹²-6x¹⁰+104x⁶+93x⁴+18x²+4	2³⁴·3⁴⁰·47⁴
12T64	x¹²-x⁸+9x⁴-1	-2⁴⁰·13⁸
12T65	x¹²-24x¹⁰+221x⁸-976x⁶+2108x⁴-2016x²+676	2⁴⁶·7⁶·13²·17⁸

Matrix representation of C₄²⋊S₃ ►in GL₃(𝔽₅) generated by

1	0	2
3	2	4
1	0	0

0	1	0
2	1	0
4	3	2

4	3	2
3	0	2
0	0	1

1	2	3
0	4	0
0	0	4

G:=sub<GL(3,GF(5))| [1,3,1,0,2,0,2,4,0],[0,2,4,1,1,3,0,0,2],[4,3,0,3,0,0,2,2,1],[1,0,0,2,4,0,3,0,4] >;

C₄²⋊S₃ in GAP, Magma, Sage, TeX

C_4^2\rtimes S_3

% in TeX

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

// GroupNames label

G:=SmallGroup(96,64);

// by ID

G=gap.SmallGroup(96,64);

# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,49,218,116,230,147,801,69,2164,730,1307]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₄²⋊S₃ in TeX
Character table of C₄²⋊S₃ in TeX

G = C42⋊S3 order 96 = 25·3

The semidirect product of C42 and S3 acting faithfully

G = C₄²⋊S₃ order 96 = 2⁵·3

The semidirect product of C₄² and S₃ acting faithfully