Copied to
clipboard

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(S3×D4), C3⋊3(D4×D9), (C2×C6)⋊5D18, (C2×C18)⋊5D6, C9⋊D4⋊2S3, C3⋊D4⋊2D9, (S3×C6).8D6, C22⋊5(S3×D9), C9⋊D12⋊6C2, C3⋊D36⋊6C2, (C6×C18)⋊4C22, (C6×D9)⋊4C22, C32.5(S3×D4), (S3×C18)⋊4C22, C18.D6⋊3C2, (C3×Dic3).8D6, C6.27(C22×D9), C3.1(Dic3⋊D6), (C3×C18).27C23, C18.27(C22×S3), (C3×Dic9)⋊2C22, (C9×Dic3)⋊2C22, (C2×C6).7S32, (C2×S3×D9)⋊6C2, (C3×C9)⋊9(C2×D4), C6.46(C2×S32), C2.27(C2×S3×D9), (C3×C9⋊D4)⋊4C2, (C9×C3⋊D4)⋊4C2, (C2×C9⋊S3)⋊7C22, (C22×C9⋊S3)⋊3C2, (C3×C3⋊D4).4S3, (C3×C6).95(C22×S3), SmallGroup(432,315)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C18 — D18⋊D6
Chief series
C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — C2×S3×D9 — D18⋊D6
Lower central
C3×C9 — C3×C18 — 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, C2×C4, D4, C23, C9, C9, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3⋊D4, C3×D4, C22×S3, C3×C9, Dic9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, C3×Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×D4, C3×D9, S3×C9, C9⋊S3, C9⋊S3, C3×C18, C3×C18, C4×D9, D36, C9⋊D4, C9⋊D4, D4×C9, C22×D9, C6.D6, C3⋊D12, C3×C3⋊D4, C3×C3⋊D4, C2×S32, C22×C3⋊S3, C3×Dic9, C9×Dic3, S3×D9, C6×D9, S3×C18, C2×C9⋊S3, C2×C9⋊S3, C6×C18, D4×D9, Dic3⋊D6, C18.D6, C3⋊D36, C9⋊D12, C3×C9⋊D4, C9×C3⋊D4, C2×S3×D9, C22×C9⋊S3, D18⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, D18, S32, S3×D4, C22×D9, C2×S32, S3×D9, D4×D9, Dic3⋊D6, C2×S3×D9, 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++++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6D6D6D6D9D18D18D18S32S3×D4S3×D4C2×S32S3×D9D4×D9Dic3⋊D6C2×S3×D9D18⋊D6
kernelD18⋊D6C18.D6C3⋊D36C9⋊D12C3×C9⋊D4C9×C3⋊D4C2×S3×D9C22×C9⋊S3C9⋊D4C3×C3⋊D4C9⋊S3Dic9D18C2×C18C3×Dic3S3×C6C62C3⋊D4Dic3D6C2×C6C2×C6C9C32C6C22C3C3C2C1
# 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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁