Copied to
clipboard

G = Dic6:D6order 288 = 25·32

4th semidirect product of Dic6 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: Dic6:4D6, C3:C8:15D6, D4.7S32, D4.S3:2S3, C3:S3:4SD16, (C3xD4).9D6, C3:3(S3xSD16), C6.57(S3xD4), C3:Dic3.20D4, (C3xC12).7C23, C12.7(C22xS3), C32:10(C2xSD16), Dic3.D6:3C2, C12.29D6:4C2, (C3xDic6):7C22, C32:5SD16:11C2, C2.17(Dic3:D6), C12:S3.7C22, (D4xC32).3C22, C4.7(C2xS32), (D4xC3:S3).2C2, (C3xC3:C8):15C22, (C3xD4.S3):6C2, (C2xC3:S3).57D4, (C3xC6).122(C2xD4), (C4xC3:S3).13C22, SmallGroup(288,578)

Series: Derived Chief Lower central Upper central

C1C3xC12 — Dic6:D6
C1C3C32C3xC6C3xC12C3xDic6Dic3.D6 — Dic6:D6
C32C3xC6C3xC12 — Dic6:D6
C1C2C4D4

Generators and relations for Dic6:D6
 G = < a,b,c,d | a12=c6=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a7, cbc-1=dbd=a9b, dcd=c-1 >

Subgroups: 834 in 163 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC8, SD16, C2xD4, C2xQ8, C3:S3, C3:S3, C3xC6, C3xC6, C3:C8, C24, Dic6, Dic6, C4xS3, D12, C3:D4, C3xD4, C3xD4, C3xQ8, C22xS3, C2xSD16, C3xDic3, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C62, S3xC8, C24:C2, D4.S3, Q8:2S3, C3xSD16, S3xD4, S3xQ8, C3xC3:C8, C6.D6, C32:2Q8, C3xDic6, C4xC3:S3, C12:S3, C32:7D4, D4xC32, C22xC3:S3, S3xSD16, C12.29D6, C32:5SD16, C3xD4.S3, Dic3.D6, D4xC3:S3, Dic6:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C22xS3, C2xSD16, S32, S3xD4, C2xS32, S3xSD16, Dic3:D6, Dic6:D6

Permutation representations of Dic6:D6
On 24 points - transitive group 24T670
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 13 7 19)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)
(1 5 9)(2 12 10 8 6 4)(3 7 11)(13 24 17 16 21 20)(14 19 18 23 22 15)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)

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

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

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

G:=TransitiveGroup(24,670);

33 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E24A24B24C24D
order122222333444466666668888121212121224242424
size1149936224212121822488886666448242412121212

33 irreducible representations

dim1111112222222444448
type+++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6SD16S32S3xD4C2xS32S3xSD16Dic3:D6Dic6:D6
kernelDic6:D6C12.29D6C32:5SD16C3xD4.S3Dic3.D6D4xC3:S3D4.S3C3:Dic3C2xC3:S3C3:C8Dic6C3xD4C3:S3D4C6C4C3C2C1
# reps1122112112224121421

Matrix representation of Dic6:D6 in GL6(F73)

72480000
310000
001000
000100
0000072
0000172
,
1240000
55610000
0072000
0007200
000001
000010
,
100000
70720000
0017200
001000
000010
000001
,
7200000
310000
001000
0017200
000001
000010

G:=sub<GL(6,GF(73))| [72,3,0,0,0,0,48,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[12,55,0,0,0,0,4,61,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,70,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,3,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic6:D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes D_6
% in TeX

G:=Group("Dic6:D6");
// GroupNames label

G:=SmallGroup(288,578);
// by ID

G=gap.SmallGroup(288,578);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^6=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<