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G = D6⋊D18  order 432 = 24·33

4th semidirect product of D6 and D18 acting via D18/D9=C2

metabelian, supersoluble, monomial

Aliases: D6⋊4D18, D18⋊4D6, Dic9⋊2D6, Dic3⋊2D18, C62.72D6, C9⋊S3⋊2D4, C9⋊3(S3×D4), C3⋊3(D4×D9), (C2×C18)⋊5D6, (C2×C6)⋊5D18, 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), C18.27(C22×S3), (C3×C18).27C23, (C9×Dic3)⋊2C22, (C3×Dic9)⋊2C22, (C2×S3×D9)⋊6C2, (C3×C9)⋊9(C2×D4), C6.46(C2×S32), C2.27(C2×S3×D9), (C2×C6).7(S32), (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 — D6⋊D18
Chief series
C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — C2×S3×D9 — D6⋊D18
Lower central
C3×C9 — C3×C18 — D6⋊D18
Upper central
C1 — C2 — C22

Generators and relations for D6⋊D18
 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 [×6], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×12], C6 [×2], C6 [×7], C2×C4, D4 [×4], C23 [×2], C9, C9, C32, Dic3, Dic3, C12 [×2], D6, D6 [×19], C2×C6 [×2], C2×C6 [×3], C2×D4, D9 [×8], C18, C18 [×5], C3×S3 [×2], C3⋊S3 [×3], C3×C6, C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4, C3⋊D4 [×3], C3×D4 [×2], C22×S3 [×5], C3×C9, Dic9, C36, D18, D18 [×12], C2×C18, C2×C18 [×2], C3×Dic3, C3×Dic3, S32 [×2], S3×C6, S3×C6, C2×C3⋊S3 [×4], C62, S3×D4 [×2], C3×D9, S3×C9, C9⋊S3 [×2], C9⋊S3, C3×C18, C3×C18, C4×D9, D36, C9⋊D4, C9⋊D4, D4×C9, C22×D9 [×3], C6.D6, C3⋊D12 [×2], C3×C3⋊D4, C3×C3⋊D4, C2×S32, C22×C3⋊S3, C3×Dic9, C9×Dic3, S3×D9 [×2], C6×D9, S3×C18, C2×C9⋊S3 [×2], C2×C9⋊S3 [×2], C6×C18, D4×D9, Dic3⋊D6, C18.D6, C3⋊D36, C9⋊D12, C3×C9⋊D4, C9×C3⋊D4, C2×S3×D9, C22×C9⋊S3, D6⋊D18
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D9, C22×S3 [×2], D18 [×3], S32, S3×D4 [×2], C22×D9, C2×S32, S3×D9, D4×D9, Dic3⋊D6, C2×S3×D9, D6⋊D18

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

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

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

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×D9D6⋊D18
kernelD6⋊D18C18.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 D6⋊D18 ►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] >;
 

D6⋊D18 in GAP, Magma, Sage, TeX 

D_6\rtimes D_{18}
 
% in TeX
 
G:=Group("D6:D18");
 
// 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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