Copied to
clipboard

G = D36⋊C3  order 216 = 23·33

The semidirect product of D36 and C3 acting faithfully

metacyclic, supersoluble, monomial

Aliases: D36⋊C3, C36⋊1C6, D18⋊1C6, C32.D12, 3- 1+2⋊1D4, C4⋊(C9⋊C6), C9⋊1(C3×D4), C6.14(S3×C6), C12.6(C3×S3), (C3×C12).3S3, C18.3(C2×C6), C3.3(C3×D12), (C3×C6).11D6, (C4×3- 1+2)⋊1C2, (C2×3- 1+2).3C22, (C2×C9⋊C6)⋊1C2, C2.4(C2×C9⋊C6), SmallGroup(216,54)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — D36⋊C3
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — D36⋊C3
Lower central
C9 — C18 — D36⋊C3
Upper central
C1 — C2 — C4

Generators and relations for D36⋊C3
 G = < a,b,c | a36=b2=c3=1, bab=a-1, cac-1=a13, cbc-1=a12b >

C1
C2
18C2
18C2
C3
3C3
C4
9C22
9C22
C6
3C6
6S3
6S3
18C6
18C6
C9
C32
2C9
9D4
C12
3C12
3D6
3D6
9C2×C6
9C2×C6
C3×C6
C18
2C18
2D9
2D9
6C3×S3
6C3×S3
3- 1+2
3D12
9C3×D4
C3×C12
D18
C36
D18
2C36
3S3×C6
3S3×C6
C2×3- 1+2
2C9⋊C6
2C9⋊C6
D36
3C3×D12
C2×C9⋊C6
C2×C9⋊C6
C4×3- 1+2
D36⋊C3

Smallest permutation representation of D36⋊C3
►On 36 points
Generators in S36
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)

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

D36⋊C3 is a maximal subgroup of   C72⋊2C6  D72⋊C3  D36⋊C6  D36.C6  D36⋊6C6  D4×C9⋊C6  D36⋊3C6
D36⋊C3 is a maximal quotient of   C72.C6  C72⋊2C6  D72⋊C3  C36⋊C12  D18⋊C12

31 conjugacy classes

class 1 2A2B2C3A3B3C 4 6A6B6C6D6E6F6G9A9B9C12A12B12C12D18A18B18C36A···36F
order1222333466666669991212121218181836···36
size11181823322331818181866622666666···6

31 irreducible representations

dim11111122222222666
type++++++++++
imageC1C2C2C3C6C6S3D4D6C3×S3C3×D4D12S3×C6C3×D12C9⋊C6C2×C9⋊C6D36⋊C3
kernelD36⋊C3C4×3- 1+2C2×C9⋊C6D36C36D18C3×C123- 1+2C3×C6C12C9C32C6C3C4C2C1
# reps11222411122224112

Matrix representation of D36⋊C3 ►in GL8(𝔽37)

275000000
3232000000
00000001
0000003636
00100000
00010000
00001000
00000100
,
361000000
01000000
00000001
00000010
00000100
00001000
00010000
00100000
,
260000000
026000000
00100000
00010000
0000363600
00001000
00000001
0000003636

G:=sub<GL(8,GF(37))| [27,32,0,0,0,0,0,0,5,32,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0],[36,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36] >;
 

D36⋊C3 in GAP, Magma, Sage, TeX 

D_{36}\rtimes C_3
 
% in TeX
 
G:=Group("D36:C3");
 
// GroupNames label
 
G:=SmallGroup(216,54);
 
// by ID
 
G=gap.SmallGroup(216,54);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,3604,736,208,5189]);
 
// Polycyclic
 
G:=Group<a,b,c|a^36=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;
 
// generators/relations
 

Export

Subgroup lattice of D36⋊C3 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁