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## G = Dic9⋊C6order 216 = 23·33

### The semidirect product of Dic9 and C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — Dic9⋊C6
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — Dic9⋊C6
 Lower central C9 — C18 — Dic9⋊C6
 Upper central C1 — C2 — C22

Generators and relations for Dic9⋊C6
G = < a,b,c | a18=c6=1, b2=a9, bab-1=a-1, cac-1=a7, cbc-1=a9b >

Smallest permutation representation of Dic9⋊C6
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 20 10 29)(2 19 11 28)(3 36 12 27)(4 35 13 26)(5 34 14 25)(6 33 15 24)(7 32 16 23)(8 31 17 22)(9 30 18 21)
(1 20)(2 33 8 21 14 27)(3 28 15 22 9 34)(4 23)(5 36 11 24 17 30)(6 31 18 25 12 19)(7 26)(10 29)(13 32)(16 35)

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

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

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

Dic9⋊C6 is a maximal subgroup of   D366C6  D4×C9⋊C6  Dic182C6
Dic9⋊C6 is a maximal quotient of   Dic9⋊C12  D18⋊C12  Dic18⋊C6  D36⋊C6  Dic18.C6  D36.C6  C62.27D6

31 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A 9B 9C 12A 12B 18A ··· 18I order 1 2 2 2 3 3 3 4 6 6 6 6 6 6 6 6 6 9 9 9 12 12 18 ··· 18 size 1 1 2 18 2 3 3 18 2 2 2 3 3 6 6 18 18 6 6 6 18 18 6 ··· 6

31 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 C3×S3 C3×D4 C3⋊D4 S3×C6 C3×C3⋊D4 C9⋊C6 C2×C9⋊C6 Dic9⋊C6 kernel Dic9⋊C6 C9⋊C12 C2×C9⋊C6 C22×3- 1+2 C9⋊D4 Dic9 D18 C2×C18 C62 3- 1+2 C3×C6 C2×C6 C9 C32 C6 C3 C22 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 1 1 2

Matrix representation of Dic9⋊C6 in GL6(𝔽37)

 0 0 1 1 0 0 0 0 36 0 0 0 0 0 0 0 1 1 0 0 0 0 36 0 0 36 0 0 0 0 1 1 0 0 0 0
,
 30 23 0 0 0 0 30 7 0 0 0 0 0 0 0 0 14 7 0 0 0 0 30 23 0 0 14 7 0 0 0 0 30 23 0 0
,
 7 14 0 0 0 0 23 30 0 0 0 0 0 0 23 30 0 0 0 0 7 30 0 0 0 0 0 0 7 30 0 0 0 0 7 14

G:=sub<GL(6,GF(37))| [0,0,0,0,0,1,0,0,0,0,36,1,1,36,0,0,0,0,1,0,0,0,0,0,0,0,1,36,0,0,0,0,1,0,0,0],[30,30,0,0,0,0,23,7,0,0,0,0,0,0,0,0,14,30,0,0,0,0,7,23,0,0,14,30,0,0,0,0,7,23,0,0],[7,23,0,0,0,0,14,30,0,0,0,0,0,0,23,7,0,0,0,0,30,30,0,0,0,0,0,0,7,7,0,0,0,0,30,14] >;

Dic9⋊C6 in GAP, Magma, Sage, TeX

{\rm Dic}_9\rtimes C_6
% in TeX

G:=Group("Dic9:C6");
// GroupNames label

G:=SmallGroup(216,62);
// by ID

G=gap.SmallGroup(216,62);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,3604,736,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^18=c^6=1,b^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations

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