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## G = C42⋊Dic3order 192 = 26·3

### The semidirect product of C42 and Dic3 acting faithfully

Aliases: C42⋊Dic3, C23.2S4, C42⋊C3⋊C4, C41D4.S3, C23.A4.C2, C22.3(A4⋊C4), SmallGroup(192,185)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊C3 — C42⋊Dic3
 Chief series C1 — C22 — C42 — C42⋊C3 — C23.A4 — C42⋊Dic3
 Lower central C42⋊C3 — C42⋊Dic3
 Upper central C1

Generators and relations for C42⋊Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, cac-1=a-1b, dad-1=b, cbc-1=ab2, dbd-1=a-1, dcd-1=c-1 >

3C2
4C2
12C2
16C3
6C22
6C4
6C22
12C22
24C4
24C4
16C6
3C23
12D4
12C2×C4
12C2×C4
12D4
4A4
16Dic3

Character table of C42⋊Dic3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6 size 1 3 4 12 32 12 24 24 24 24 32 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 1 -1 -1 1 1 i -i i -i -1 linear of order 4 ρ4 1 1 -1 -1 1 1 -i i -i i -1 linear of order 4 ρ5 2 2 2 2 -1 2 0 0 0 0 -1 orthogonal lifted from S3 ρ6 2 2 -2 -2 -1 2 0 0 0 0 1 symplectic lifted from Dic3, Schur index 2 ρ7 3 3 3 -1 0 -1 1 1 -1 -1 0 orthogonal lifted from S4 ρ8 3 3 3 -1 0 -1 -1 -1 1 1 0 orthogonal lifted from S4 ρ9 3 3 -3 1 0 -1 -i i i -i 0 complex lifted from A4⋊C4 ρ10 3 3 -3 1 0 -1 i -i -i i 0 complex lifted from A4⋊C4 ρ11 12 -4 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C42⋊Dic3
On 16 points - transitive group 16T430
Generators in S16
(1 16 3 13)(2 6 4 9)(5 7 12 11)(8 14 15 10)
(1 10 4 7)(2 11 3 14)(5 13 15 6)(8 9 12 16)
(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
(3 4)(5 15 8 12)(6 14 9 11)(7 13 10 16)

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

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

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

G:=TransitiveGroup(16,430);

On 24 points - transitive group 24T374
Generators in S24
(1 13 4 16)(2 21 5 24)(3 10)(6 9)(7 23 11 20)(8 17 12 14)(15 22)(18 19)
(1 7)(2 14 5 17)(3 19 6 22)(4 11)(8 24 12 21)(9 15 10 18)(13 23)(16 20)
(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)(2 5)(3 4)(7 9)(10 11)(13 19 16 22)(14 24 17 21)(15 23 18 20)

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

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

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

G:=TransitiveGroup(24,374);

On 24 points - transitive group 24T378
Generators in S24
(1 8)(2 24 9 21)(3 22 7 19)(4 16 10 13)(5 17 11 14)(20 23)
(1 20 8 23)(3 19 7 22)(4 10)(5 17 11 14)(6 15 12 18)(13 16)
(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 10)(2 12)(3 11)(4 8)(5 7)(6 9)(13 23 16 20)(14 22 17 19)(15 21 18 24)

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

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

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

G:=TransitiveGroup(24,378);

On 24 points - transitive group 24T384
Generators in S24
(1 8 23 15)(2 21)(3 14 19 7)(4 18 20 11)(5 24)(6 10 22 17)(9 12)(13 16)
(1 11 23 18)(2 16 24 9)(3 6)(4 15 20 8)(5 12 21 13)(7 17)(10 14)(19 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)
(1 18 4 15)(2 17 5 14)(3 16 6 13)(7 24 10 21)(8 23 11 20)(9 22 12 19)

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

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

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

G:=TransitiveGroup(24,384);

On 24 points - transitive group 24T387
Generators in S24
(2 16 13 5)(3 17 14 6)(7 10 20 23)(8 24 21 11)(9 22)(12 19)
(1 15 18 4)(2 13)(3 17 14 6)(5 16)(8 11 21 24)(9 12 22 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)
(1 10 4 7)(2 9 5 12)(3 8 6 11)(13 22 16 19)(14 21 17 24)(15 20 18 23)

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

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

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

G:=TransitiveGroup(24,387);

Matrix representation of C42⋊Dic3 in GL12(ℤ)

 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
,
 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0

G:=sub<GL(12,Integers())| [0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0] >;

C42⋊Dic3 in GAP, Magma, Sage, TeX

C_4^2\rtimes {\rm Dic}_3
% in TeX

G:=Group("C4^2:Dic3");
// GroupNames label

G:=SmallGroup(192,185);
// by ID

G=gap.SmallGroup(192,185);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,14,170,675,2194,857,360,5464,1271,1593,102,6053,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a^-1*b,d*a*d^-1=b,c*b*c^-1=a*b^2,d*b*d^-1=a^-1,d*c*d^-1=c^-1>;
// generators/relations

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