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G = D18:D6  order 432 = 24·33

4th semidirect product of D18 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D18:4D6, D6:4D18, Dic9:2D6, Dic3:2D18, C62.72D6, C9:S3:2D4, C9:3(S3xD4), C3:3(D4xD9), (C2xC6):5D18, (C2xC18):5D6, C9:D4:2S3, C3:D4:2D9, (S3xC6).8D6, C22:5(S3xD9), C9:D12:6C2, C3:D36:6C2, (C6xC18):4C22, (C6xD9):4C22, C32.5(S3xD4), (S3xC18):4C22, C18.D6:3C2, (C3xDic3).8D6, C6.27(C22xD9), C3.1(Dic3:D6), (C3xC18).27C23, C18.27(C22xS3), (C3xDic9):2C22, (C9xDic3):2C22, (C2xC6).7S32, (C2xS3xD9):6C2, (C3xC9):9(C2xD4), C6.46(C2xS32), C2.27(C2xS3xD9), (C3xC9:D4):4C2, (C9xC3:D4):4C2, (C2xC9:S3):7C22, (C22xC9:S3):3C2, (C3xC3:D4).4S3, (C3xC6).95(C22xS3), SmallGroup(432,315)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC18 — D18:D6
Chief series
C1 — C3 — C32 — C3xC9 — C3xC18 — S3xC18 — C2xS3xD9 — D18:D6
Lower central
C3xC9 — C3xC18 — D18:D6
Upper central
C1 — C2 — C22

Generators and relations for D18:D6
 G = < a,b,c,d | a6=b2=c18=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 1544 in 194 conjugacy classes, 43 normal (41 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C9, C9, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xD4, D9, C18, C18, C3xS3, C3:S3, C3xC6, C3xC6, C4xS3, D12, C3:D4, C3:D4, C3xD4, C22xS3, C3xC9, Dic9, C36, D18, D18, C2xC18, C2xC18, C3xDic3, C3xDic3, S32, S3xC6, S3xC6, C2xC3:S3, C62, S3xD4, C3xD9, S3xC9, C9:S3, C9:S3, C3xC18, C3xC18, C4xD9, D36, C9:D4, C9:D4, D4xC9, C22xD9, C6.D6, C3:D12, C3xC3:D4, C3xC3:D4, C2xS32, C22xC3:S3, C3xDic9, C9xDic3, S3xD9, C6xD9, S3xC18, C2xC9:S3, C2xC9:S3, C6xC18, D4xD9, Dic3:D6, C18.D6, C3:D36, C9:D12, C3xC9:D4, C9xC3:D4, C2xS3xD9, C22xC9:S3, D18:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D9, C22xS3, D18, S32, S3xD4, C22xD9, C2xS32, S3xD9, D4xD9, Dic3:D6, C2xS3xD9, D18:D6

Smallest permutation representation of D18:D6
►On 36 points
Generators in S36
(1 13 4 16 7 10)(2 14 5 17 8 11)(3 15 6 18 9 12)(19 22 25 28 31 34)(20 23 26 29 32 35)(21 24 27 30 33 36)

(1 24)(2 34)(3 26)(4 36)(5 28)(6 20)(7 30)(8 22)(9 32)(10 27)(11 19)(12 29)(13 21)(14 31)(15 23)(16 33)(17 25)(18 35)

(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)

(1 9)(2 8)(3 7)(4 6)(10 12)(13 18)(14 17)(15 16)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(35 36)
 
G:=sub<Sym(36)| (1,13,4,16,7,10)(2,14,5,17,8,11)(3,15,6,18,9,12)(19,22,25,28,31,34)(20,23,26,29,32,35)(21,24,27,30,33,36), (1,24)(2,34)(3,26)(4,36)(5,28)(6,20)(7,30)(8,22)(9,32)(10,27)(11,19)(12,29)(13,21)(14,31)(15,23)(16,33)(17,25)(18,35), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36), (1,9)(2,8)(3,7)(4,6)(10,12)(13,18)(14,17)(15,16)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,36)>;
 
G:=Group( (1,13,4,16,7,10)(2,14,5,17,8,11)(3,15,6,18,9,12)(19,22,25,28,31,34)(20,23,26,29,32,35)(21,24,27,30,33,36), (1,24)(2,34)(3,26)(4,36)(5,28)(6,20)(7,30)(8,22)(9,32)(10,27)(11,19)(12,29)(13,21)(14,31)(15,23)(16,33)(17,25)(18,35), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36), (1,9)(2,8)(3,7)(4,6)(10,12)(13,18)(14,17)(15,16)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,36) );
 
G=PermutationGroup([[(1,13,4,16,7,10),(2,14,5,17,8,11),(3,15,6,18,9,12),(19,22,25,28,31,34),(20,23,26,29,32,35),(21,24,27,30,33,36)], [(1,24),(2,34),(3,26),(4,36),(5,28),(6,20),(7,30),(8,22),(9,32),(10,27),(11,19),(12,29),(13,21),(14,31),(15,23),(16,33),(17,25),(18,35)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,18),(14,17),(15,16),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(35,36)]])
 

51 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A6B6C···6G6H6I9A9B9C9D9E9F12A12B18A18B18C18D···18O18P18Q18R36A36B36C
order1222222233344666···666999999121218181818···18181818363636
size112618272754224618224···4123622244412362224···4121212121212

51 irreducible representations

dim111111112222222222222444444444
type++++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6D6D6D6D9D18D18D18S32S3xD4S3xD4C2xS32S3xD9D4xD9Dic3:D6C2xS3xD9D18:D6
kernelD18:D6C18.D6C3:D36C9:D12C3xC9:D4C9xC3:D4C2xS3xD9C22xC9:S3C9:D4C3xC3:D4C9:S3Dic9D18C2xC18C3xDic3S3xC6C62C3:D4Dic3D6C2xC6C2xC6C9C32C6C22C3C3C2C1
# reps111111111121111113333111133236

Matrix representation of D18:D6 ►in GL6(F37)

3610000
3600000
0036000
0003600
000010
000001
,
100000
1360000
0003600
0036000
0000360
0000036
,
3600000
0360000
001000
0003600
00003124
00001520
,
0360000
3600000
001000
0003600
0000613
0000331

G:=sub<GL(6,GF(37))| [36,36,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,31,15,0,0,0,0,24,20],[0,36,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,6,3,0,0,0,0,13,31] >;
 

D18:D6 in GAP, Magma, Sage, TeX 

D_{18}\rtimes D_6
 
% in TeX
 
G:=Group("D18:D6");
 
// GroupNames label
 
G:=SmallGroup(432,315);
 
// by ID
 
G=gap.SmallGroup(432,315);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,3091,662,4037,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^18=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

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