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## G = C2×C32⋊2D9order 324 = 22·34

### Direct product of C2 and C32⋊2D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — C2×C32⋊2D9
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32⋊2D9 — C2×C32⋊2D9
 Lower central C32⋊C9 — C2×C32⋊2D9
 Upper central C1 — C6

Generators and relations for C2×C322D9
G = < a,b,c,d,e | a2=b3=c3=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1c, cd=dc, ce=ec, ede=d-1 >

Subgroups: 535 in 99 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2 [×2], C3 [×2], C3 [×4], C22, S3 [×8], C6 [×2], C6 [×6], C9 [×3], C32, C32 [×3], C32 [×2], D6 [×4], C2×C6, D9 [×6], C18 [×3], C3×S3 [×8], C3⋊S3 [×2], C3×C6, C3×C6 [×3], C3×C6 [×2], C3×C9 [×3], C33, D18 [×3], S3×C6 [×4], C2×C3⋊S3, C3×D9 [×6], C3×C18 [×3], C3×C3⋊S3 [×2], C32×C6, C32⋊C9, C6×D9 [×3], C6×C3⋊S3, C322D9 [×2], C2×C32⋊C9, C2×C322D9
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], D9 [×3], C3⋊S3, D18 [×3], C2×C3⋊S3, C9⋊S3, He3⋊C2, C2×C9⋊S3, C2×He3⋊C2, C322D9, C2×C322D9

Smallest permutation representation of C2×C322D9
On 36 points
Generators in S36
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(20 23 26)(21 27 24)(29 32 35)(30 36 33)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)
(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 30)(2 29)(3 28)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 27)(18 26)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 3 3 6 6 type + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 D9 D18 He3⋊C2 C2×He3⋊C2 C32⋊2D9 C2×C32⋊2D9 kernel C2×C32⋊2D9 C32⋊2D9 C2×C32⋊C9 C3×C18 C32×C6 C3×C9 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 2 1 3 1 3 1 9 9 4 4 2 2

Matrix representation of C2×C322D9 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 10 15 0 0 0 4 8 0 0 0 0 0 18 6 18 0 0 0 1 0 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 18 10 0 0 0 9 4 0 0 0 0 0 1 0 0 0 0 18 7 7 0 0 0 0 11
,
 10 15 0 0 0 1 9 0 0 0 0 0 18 0 0 0 0 1 12 12 0 0 7 15 7

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[10,4,0,0,0,15,8,0,0,0,0,0,18,0,1,0,0,6,1,0,0,0,18,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[18,9,0,0,0,10,4,0,0,0,0,0,1,18,0,0,0,0,7,0,0,0,0,7,11],[10,1,0,0,0,15,9,0,0,0,0,0,18,1,7,0,0,0,12,15,0,0,0,12,7] >;

C2×C322D9 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2D_9
% in TeX

G:=Group("C2xC3^2:2D9");
// GroupNames label

G:=SmallGroup(324,75);
// by ID

G=gap.SmallGroup(324,75);
# by ID

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

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

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