Copied to
clipboard

## G = C62⋊Dic3order 432 = 24·33

### The semidirect product of C62 and Dic3 acting faithfully

Aliases: C62⋊Dic3, C3⋊S3.2S4, A4⋊(C32⋊C4), (C32×A4)⋊1C4, C323(A4⋊C4), C22⋊(C33⋊C4), (A4×C3⋊S3).1C2, (C22×C3⋊S3).3S3, SmallGroup(432,743)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×A4 — C62⋊Dic3
 Chief series C1 — C22 — C62 — C32×A4 — A4×C3⋊S3 — C62⋊Dic3
 Lower central C32×A4 — C62⋊Dic3
 Upper central C1

Generators and relations for C62⋊Dic3
G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, cac-1=a-1b3, dad-1=a4b-1, cbc-1=a3b2, dbd-1=a-1b2, dcd-1=c-1 >

Subgroups: 752 in 70 conjugacy classes, 10 normal (all characteristic)
C1, C2, C3, C4, C22, C22, S3, C6, C2×C4, C23, C32, C32, Dic3, A4, A4, D6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C2×A4, C22×S3, C33, C32⋊C4, C3×A4, C2×C3⋊S3, C62, A4⋊C4, C3×C3⋊S3, S3×A4, C2×C32⋊C4, C22×C3⋊S3, C33⋊C4, C32×A4, C62⋊C4, A4×C3⋊S3, C62⋊Dic3
Quotients: C1, C2, C4, S3, Dic3, S4, C32⋊C4, A4⋊C4, C33⋊C4, C62⋊Dic3

Character table of C62⋊Dic3

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 6A 6B 6C size 1 3 9 27 4 4 8 16 16 16 16 54 54 54 54 12 12 72 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 -i i i -i 1 1 -1 linear of order 4 ρ4 1 1 -1 -1 1 1 1 1 1 1 1 i -i -i i 1 1 -1 linear of order 4 ρ5 2 2 2 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 2 2 -1 orthogonal lifted from S3 ρ6 2 2 -2 -2 2 2 -1 -1 -1 -1 -1 0 0 0 0 2 2 1 symplectic lifted from Dic3, Schur index 2 ρ7 3 -1 3 -1 3 3 0 0 0 0 0 -1 1 -1 1 -1 -1 0 orthogonal lifted from S4 ρ8 3 -1 3 -1 3 3 0 0 0 0 0 1 -1 1 -1 -1 -1 0 orthogonal lifted from S4 ρ9 3 -1 -3 1 3 3 0 0 0 0 0 i i -i -i -1 -1 0 complex lifted from A4⋊C4 ρ10 3 -1 -3 1 3 3 0 0 0 0 0 -i -i i i -1 -1 0 complex lifted from A4⋊C4 ρ11 4 4 0 0 1 -2 4 -2 1 -2 1 0 0 0 0 1 -2 0 orthogonal lifted from C32⋊C4 ρ12 4 4 0 0 -2 1 4 1 -2 1 -2 0 0 0 0 -2 1 0 orthogonal lifted from C32⋊C4 ρ13 4 4 0 0 1 -2 -2 1 -1+3√-3/2 1 -1-3√-3/2 0 0 0 0 1 -2 0 complex lifted from C33⋊C4 ρ14 4 4 0 0 -2 1 -2 -1-3√-3/2 1 -1+3√-3/2 1 0 0 0 0 -2 1 0 complex lifted from C33⋊C4 ρ15 4 4 0 0 1 -2 -2 1 -1-3√-3/2 1 -1+3√-3/2 0 0 0 0 1 -2 0 complex lifted from C33⋊C4 ρ16 4 4 0 0 -2 1 -2 -1+3√-3/2 1 -1-3√-3/2 1 0 0 0 0 -2 1 0 complex lifted from C33⋊C4 ρ17 12 -4 0 0 3 -6 0 0 0 0 0 0 0 0 0 -1 2 0 orthogonal faithful ρ18 12 -4 0 0 -6 3 0 0 0 0 0 0 0 0 0 2 -1 0 orthogonal faithful

Permutation representations of C62⋊Dic3
On 24 points - transitive group 24T1339
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 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 22 11 20 9 24)(8 23 12 21 10 19)
(2 13 18 6 15 16)(3 5)(4 17 14)(7 24 10 11 20 8)(9 22 12)(21 23)
(1 19)(2 10 6 8)(3 21 5 23)(4 12)(7 16 11 18)(9 14)(13 24 15 20)(17 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), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,22,11,20,9,24)(8,23,12,21,10,19), (2,13,18,6,15,16)(3,5)(4,17,14)(7,24,10,11,20,8)(9,22,12)(21,23), (1,19)(2,10,6,8)(3,21,5,23)(4,12)(7,16,11,18)(9,14)(13,24,15,20)(17,22)>;

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

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

G:=TransitiveGroup(24,1339);

Matrix representation of C62⋊Dic3 in GL7(𝔽13)

 0 12 1 0 0 0 0 0 12 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 12 0 0 0 0 0 0 4 1 4 0 0 0 12 12 9 11
,
 12 0 0 0 0 0 0 12 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 9 9 11 9 0 0 0 1 1 4 1
,
 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 12 9 12
,
 0 8 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 12 12 9 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(7,GF(13))| [0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,12,12,4,12,0,0,0,0,0,1,9,0,0,0,0,0,4,11],[12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,9,1,0,0,0,0,1,9,1,0,0,0,0,0,11,4,0,0,0,0,0,9,1],[0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,1,0,0,12,0,0,0,0,0,1,9,0,0,0,0,0,0,12],[0,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,1,0,0,0,0,9,1,0,0,0,0,1,12,0,0] >;

C62⋊Dic3 in GAP, Magma, Sage, TeX

C_6^2\rtimes {\rm Dic}_3
% in TeX

G:=Group("C6^2:Dic3");
// GroupNames label

G:=SmallGroup(432,743);
// by ID

G=gap.SmallGroup(432,743);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,3,-2,2,14,170,1683,346,1684,1271,9077,2287,5298,3989]);
// Polycyclic

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

Export

׿
×
𝔽