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G = C42⋊Dic3  order 192 = 26·3

The semidirect product of C42 and Dic3 acting faithfully

non-abelian, soluble, monomial

Aliases: C42⋊Dic3, C23.2S4, C42⋊C3⋊C4, C4⋊1D4.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 >

C1
3C2
4C2
12C2
16C3
C22
6C22
6C4
6C22
12C22
24C4
24C4
16C6
C23
3C2×C4
3C23
12D4
12C2×C4
12C2×C4
12D4
4A4
16Dic3
C42
3C2×D4
3C2×D4
6C22⋊C4
6C22⋊C4
4C2×A4
C4⋊1D4
3C23⋊C4
3C23⋊C4
C42⋊C3
4A4⋊C4
3C42⋊C4
C23.A4
C42⋊Dic3

Character table of C42⋊Dic3

 class 12A2B2C34A4B4C4D4E6
 size 1341232122424242432
ρ111111111111    trivial
ρ2111111-1-1-1-11    linear of order 2
ρ311-1-111i-ii-i-1    linear of order 4
ρ411-1-111-ii-ii-1    linear of order 4
ρ52222-120000-1    orthogonal lifted from S3
ρ622-2-2-1200001    symplectic lifted from Dic3, Schur index 2
ρ7333-10-111-1-10    orthogonal lifted from S4
ρ8333-10-1-1-1110    orthogonal lifted from S4
ρ933-310-1-iii-i0    complex lifted from A4⋊C4
ρ1033-310-1i-i-ii0    complex lifted from A4⋊C4
ρ1112-4000000000    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)

000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
-1-1-1000000000
001000000000
010000000000
000-1-1-1000000
000001000000
000010000000
,
000001000000
000-1-1-1000000
000100000000
100000000000
010000000000
001000000000
000000000001
000000000-1-1-1
000000000100
000000100000
000000010000
000000001000
,
100000000000
001000000000
-1-1-1000000000
000000010000
000000-1-1-1000
000000001000
000000000010
000000000-1-1-1
000000000001
000-1-1-1000000
000010000000
000100000000
,
100000000000
010000000000
-1-1-1000000000
000000100000
000000010000
000000-1-1-1000
000001000000
000-1-1-1000000
000010000000
000000000001
000000000-1-1-1
000000000010

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
 

Export

Subgroup lattice of C42⋊Dic3 in TeX
Character table of C42⋊Dic3 in TeX

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