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## G = C22×C9⋊C6order 216 = 23·33

### Direct product of C22 and C9⋊C6

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×C9⋊C6
G = < a,b,c,d | a2=b2=c9=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 336 in 101 conjugacy classes, 47 normal (12 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C9, C9, C32, D6, C2×C6, C2×C6, D9, C18, C18, C3×S3, C3×C6, C22×S3, C22×C6, 3- 1+2, D18, C2×C18, C2×C18, S3×C6, C62, C9⋊C6, C2×3- 1+2, C22×D9, S3×C2×C6, C2×C9⋊C6, C22×3- 1+2, C22×C9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C3×S3, C22×S3, C22×C6, S3×C6, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C22×C9⋊C6

Smallest permutation representation of C22×C9⋊C6
On 36 points
Generators in S36
(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 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 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 29)(2 34 8 28 5 31)(3 30 6 36 9 33)(4 35)(7 32)(10 24 13 21 16 27)(11 20)(12 25 18 19 15 22)(14 26)(17 23)

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

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

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

C22×C9⋊C6 is a maximal subgroup of   D18⋊C12
C22×C9⋊C6 is a maximal quotient of   D366C6  Dic182C6  D363C6

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D ··· 6I 6J ··· 6Q 9A 9B 9C 18A ··· 18I order 1 2 2 2 2 2 2 2 3 3 3 6 6 6 6 ··· 6 6 ··· 6 9 9 9 18 ··· 18 size 1 1 1 1 9 9 9 9 2 3 3 2 2 2 3 ··· 3 9 ··· 9 6 6 6 6 ··· 6

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 6 6 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 C9⋊C6 C2×C9⋊C6 kernel C22×C9⋊C6 C2×C9⋊C6 C22×3- 1+2 C22×D9 D18 C2×C18 C62 C3×C6 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 1 3 2 6 1 3

Matrix representation of C22×C9⋊C6 in GL8(𝔽19)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18
,
 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18
,
 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 1 1 18 17 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 18 0 1 0 0 1 1 0 18 18 18 0 0 0 0 0 18 0 0 0 0 1 0 0 18 0 0
,
 0 12 0 0 0 0 0 0 12 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 0 1 0 0 0 1 1 0 0 18 18 0 0 1 1 18 18 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18],[18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18],[18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,18,1,0,0,0,0,0,0,17,18,18,18,18,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,1,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0] >;

C22×C9⋊C6 in GAP, Magma, Sage, TeX

C_2^2\times C_9\rtimes C_6
% in TeX

G:=Group("C2^2xC9:C6");
// GroupNames label

G:=SmallGroup(216,111);
// by ID

G=gap.SmallGroup(216,111);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,3604,382,208,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^9=d^6=1,a*b=b*a,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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