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G = C42⋊S3order 96 = 25·3

The semidirect product of C42 and S3 acting faithfully

non-abelian, soluble, monomial

Aliases: C42⋊S3, C22.S4, C42⋊C32C2, SmallGroup(96,64)

Series: Derived Chief Lower central Upper central

C1C42C42⋊C3 — C42⋊S3
C1C22C42C42⋊C3 — C42⋊S3
C42⋊C3 — C42⋊S3
C1

Generators and relations for C42⋊S3
 G = < a,b,c,d | a4=b4=c3=d2=1, ab=ba, cac-1=dad=b, cbc-1=a-1b-1, dbd=a, dcd=c-1 >

3C2
12C2
16C3
3C4
3C4
6C22
6C4
16S3
3C2×C4
3D4
3Q8
6D4
6C8
6C2×C4
4A4
3C4○D4
3M4(2)
4S4
3C4≀C2

Character table of C42⋊S3

 class 12A2B34A4B4C4D8A8B
 size 131232336121212
ρ11111111111    trivial
ρ211-11111-1-1-1    linear of order 2
ρ3220-1222000    orthogonal lifted from S3
ρ433-10-1-1-1-111    orthogonal lifted from S4
ρ53310-1-1-11-1-1    orthogonal lifted from S4
ρ63-110-1+2i-1-2i1-1-ii    complex faithful
ρ73-1-10-1+2i-1-2i11i-i    complex faithful
ρ83-1-10-1-2i-1+2i11-ii    complex faithful
ρ93-110-1-2i-1+2i1-1i-i    complex faithful
ρ106-20022-2000    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 6 7 8)(9 11)(10 12)
(1 8 9)(2 6 11)(3 7 12)(4 5 10)
(1 9)(2 11)(3 12)(4 10)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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
actionf(x)Disc(f)
12T62x12-13x10+59x8-109x6+73x4-16x2+1220·194·1034
12T63x12-6x10+104x6+93x4+18x2+4234·340·474
12T64x12-x8+9x4-1-240·138
12T65x12-24x10+221x8-976x6+2108x4-2016x2+676246·76·132·178

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

102
324
100
,
010
210
432
,
432
302
001
,
123
040
004
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

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