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G = D4×C9⋊C6  order 432 = 24·33

Direct product of D4 and C9⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: D4×C9⋊C6, D36⋊2C6, C62.14D6, C36⋊(C2×C6), D9⋊(C3×D4), (D4×D9)⋊C3, C9⋊2(C6×D4), (D4×C9)⋊2C6, (C4×D9)⋊1C6, C9⋊D4⋊1C6, D36⋊C3⋊2C2, C32.(S3×D4), D18⋊2(C2×C6), C12.17(S3×C6), C9⋊C12⋊1C22, Dic9⋊C6⋊1C2, Dic9⋊1(C2×C6), (C3×C12).26D6, (C22×D9)⋊2C6, C18.5(C22×C6), (D4×C32).5S3, 3- 1+2⋊2(C2×D4), (D4×3- 1+2)⋊2C2, (C4×3- 1+2)⋊C22, (C22×3- 1+2)⋊C22, (C2×3- 1+2).5C23, C4⋊1(C2×C9⋊C6), (C2×C18)⋊(C2×C6), (C4×C9⋊C6)⋊1C2, C3.3(C3×S3×D4), C6.35(S3×C2×C6), C22⋊3(C2×C9⋊C6), (C2×C9⋊C6)⋊2C22, (C22×C9⋊C6)⋊2C2, (C2×C6).12(S3×C6), C2.6(C22×C9⋊C6), (C3×D4).11(C3×S3), (C3×C6).29(C22×S3), Aut(D36), Hol(C36), SmallGroup(432,362)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — D4×C9⋊C6
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C22×C9⋊C6 — D4×C9⋊C6
Lower central
C9 — C18 — D4×C9⋊C6
Upper central
C1 — C2 — D4

Generators and relations for D4×C9⋊C6
 G = < a,b,c,d | a4=b2=c9=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 750 in 170 conjugacy classes, 56 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×D4, D9, D9, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, 3- 1+2, Dic9, C36, C36, D18, D18, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, S3×D4, C6×D4, C9⋊C6, C9⋊C6, C2×3- 1+2, C2×3- 1+2, C4×D9, D36, C9⋊D4, D4×C9, D4×C9, C22×D9, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, S3×C2×C6, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C2×C9⋊C6, C2×C9⋊C6, C22×3- 1+2, D4×D9, C3×S3×D4, C4×C9⋊C6, D36⋊C3, Dic9⋊C6, D4×3- 1+2, C22×C9⋊C6, D4×C9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, S3×C6, S3×D4, C6×D4, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C3×S3×D4, C22×C9⋊C6, D4×C9⋊C6

Smallest permutation representation of D4×C9⋊C6
►On 36 points
Generators in S36
(1 29 11 20)(2 30 12 21)(3 31 13 22)(4 32 14 23)(5 33 15 24)(6 34 16 25)(7 35 17 26)(8 36 18 27)(9 28 10 19)

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O6P6Q9A9B9C12A12B12C12D12E18A18B18C18D···18I36A36B36C
order122222223334466666666666666666999121212121218181818···18363636
size1122991818233218233446666999918181818666466181866612···12121212

50 irreducible representations

dim111111111111122222222244666
type+++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4×C9⋊C6S3D4D6D6C3×S3C3×D4S3×C6S3×C6S3×D4C3×S3×D4C9⋊C6C2×C9⋊C6C2×C9⋊C6
kernelD4×C9⋊C6C4×C9⋊C6D36⋊C3Dic9⋊C6D4×3- 1+2C22×C9⋊C6D4×D9C4×D9D36C9⋊D4D4×C9C22×D9C1D4×C32C9⋊C6C3×C12C62C3×D4D9C12C2×C6C32C3D4C4C22
# reps11121222242411212242412112

Matrix representation of D4×C9⋊C6 ►in GL10(ℤ)

0010000000
0001000000
-1000000000
0-100000000
0000-100000
00000-10000
000000-1000
0000000-100
00000000-10
000000000-1
,
0010000000
0001000000
1000000000
0100000000
0000-100000
00000-10000
000000-1000
0000000-100
00000000-10
000000000-1
,
-1100000000
-1000000000
00-11000000
00-10000000
0000000100
000000-1-100
0000000001
00000000-1-1
0000100000
0000010000
,
0-100000000
-1000000000
000-1000000
00-10000000
0000100000
0000-1-10000
0000000010
00000000-1-1
000000-1-100
0000000100

G:=sub<GL(10,Integers())| [0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1],[-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0],[0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0] >;
 

D4×C9⋊C6 in GAP, Magma, Sage, TeX 

D_4\times C_9\rtimes C_6
 
% in TeX
 
G:=Group("D4xC9:C6");
 
// GroupNames label
 
G:=SmallGroup(432,362);
 
// by ID
 
G=gap.SmallGroup(432,362);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,10085,1034,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^9=d^6=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^2>;
 
// generators/relations
 

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