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## G = C32⋊2Dic9order 324 = 22·34

### 2nd semidirect product of C32 and Dic9 acting via Dic9/C6=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — C32⋊2Dic9
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C2×C32⋊C9 — C32⋊2Dic9
 Lower central C32⋊C9 — C32⋊2Dic9
 Upper central C1 — C6

Generators and relations for C322Dic9
G = < a,b,c,d | a3=b3=c18=1, d2=c9, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 233 in 61 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C32, C32, C32, Dic3, C12, C18, C3×C6, C3×C6, C3×C6, C3×C9, C33, Dic9, C3×Dic3, C3⋊Dic3, C3×C18, C32×C6, C32⋊C9, C3×Dic9, C3×C3⋊Dic3, C2×C32⋊C9, C322Dic9
Quotients: C1, C2, C4, S3, Dic3, D9, C3⋊S3, Dic9, C3⋊Dic3, C9⋊S3, He3⋊C2, C9⋊Dic3, He33C4, C322D9, C322Dic9

Smallest permutation representation of C322Dic9
On 36 points
Generators in S36
(2 8 14)(3 15 9)(5 11 17)(6 18 12)(20 32 26)(21 27 33)(23 35 29)(24 30 36)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 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 28 10 19)(2 27 11 36)(3 26 12 35)(4 25 13 34)(5 24 14 33)(6 23 15 32)(7 22 16 31)(8 21 17 30)(9 20 18 29)

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

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

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

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 9A ··· 9I 12A 12B 12C 12D 18A ··· 18I order 1 2 3 3 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 9 ··· 9 12 12 12 12 18 ··· 18 size 1 1 1 1 2 2 2 6 6 6 27 27 1 1 2 2 2 6 6 6 6 ··· 6 27 27 27 27 6 ··· 6

42 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 3 3 6 6 type + + + + - - + - image C1 C2 C4 S3 S3 Dic3 Dic3 D9 Dic9 He3⋊C2 He3⋊3C4 C32⋊2D9 C32⋊2Dic9 kernel C32⋊2Dic9 C2×C32⋊C9 C32⋊C9 C3×C18 C32×C6 C3×C9 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 3 1 3 1 9 9 4 4 2 2

Matrix representation of C322Dic9 in GL5(𝔽37)

 14 23 0 0 0 23 22 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10
,
 15 16 0 0 0 16 27 0 0 0 0 0 36 0 0 0 0 0 27 0 0 0 0 0 11
,
 22 21 0 0 0 14 15 0 0 0 0 0 31 0 0 0 0 0 0 31 0 0 0 31 0

G:=sub<GL(5,GF(37))| [14,23,0,0,0,23,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[15,16,0,0,0,16,27,0,0,0,0,0,36,0,0,0,0,0,27,0,0,0,0,0,11],[22,14,0,0,0,21,15,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,31,0] >;

C322Dic9 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2{\rm Dic}_9
% in TeX

G:=Group("C3^2:2Dic9");
// GroupNames label

G:=SmallGroup(324,20);
// by ID

G=gap.SmallGroup(324,20);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,794,338,579,735,2164]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^18=1,d^2=c^9,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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