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## G = C2×C32⋊D9order 324 = 22·34

### Direct product of C2 and C32⋊D9

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C32⋊D9
G = < a,b,c,d,e | a2=b3=c3=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 511 in 83 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C9, C32, C32, D6, C2×C6, D9, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C9, C3×C9, C33, D18, S3×C6, C2×C3⋊S3, C9⋊S3, C3×C18, C3×C18, C3×C3⋊S3, C32×C6, C32⋊C9, C2×C9⋊S3, C6×C3⋊S3, C32⋊D9, C2×C32⋊C9, C2×C32⋊D9
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, D9, C3×S3, D18, S3×C6, C3×D9, C32⋊C6, C9⋊C6, C6×D9, C2×C32⋊C6, C2×C9⋊C6, C32⋊D9, C2×C32⋊D9

Smallest permutation representation of C2×C32⋊D9
On 54 points
Generators in S54
(1 50)(2 51)(3 52)(4 53)(5 54)(6 46)(7 47)(8 48)(9 49)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(19 44)(20 45)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)
(2 36 42)(3 43 28)(5 30 45)(6 37 31)(8 33 39)(9 40 34)(11 23 48)(12 49 24)(14 26 51)(15 52 27)(17 20 54)(18 46 21)
(1 35 41)(2 36 42)(3 28 43)(4 29 44)(5 30 45)(6 31 37)(7 32 38)(8 33 39)(9 34 40)(10 22 47)(11 23 48)(12 24 49)(13 25 50)(14 26 51)(15 27 52)(16 19 53)(17 20 54)(18 21 46)
(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(1 49)(2 48)(3 47)(4 46)(5 54)(6 53)(7 52)(8 51)(9 50)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 31)(20 30)(21 29)(22 28)(23 36)(24 35)(25 34)(26 33)(27 32)

G:=sub<Sym(54)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,46)(7,47)(8,48)(9,49)(10,32)(11,33)(12,34)(13,35)(14,36)(15,28)(16,29)(17,30)(18,31)(19,44)(20,45)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43), (2,36,42)(3,43,28)(5,30,45)(6,37,31)(8,33,39)(9,40,34)(11,23,48)(12,49,24)(14,26,51)(15,52,27)(17,20,54)(18,46,21), (1,35,41)(2,36,42)(3,28,43)(4,29,44)(5,30,45)(6,31,37)(7,32,38)(8,33,39)(9,34,40)(10,22,47)(11,23,48)(12,24,49)(13,25,50)(14,26,51)(15,27,52)(16,19,53)(17,20,54)(18,21,46), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,49)(2,48)(3,47)(4,46)(5,54)(6,53)(7,52)(8,51)(9,50)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,31)(20,30)(21,29)(22,28)(23,36)(24,35)(25,34)(26,33)(27,32)>;

G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,46)(7,47)(8,48)(9,49)(10,32)(11,33)(12,34)(13,35)(14,36)(15,28)(16,29)(17,30)(18,31)(19,44)(20,45)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43), (2,36,42)(3,43,28)(5,30,45)(6,37,31)(8,33,39)(9,40,34)(11,23,48)(12,49,24)(14,26,51)(15,52,27)(17,20,54)(18,46,21), (1,35,41)(2,36,42)(3,28,43)(4,29,44)(5,30,45)(6,31,37)(7,32,38)(8,33,39)(9,34,40)(10,22,47)(11,23,48)(12,24,49)(13,25,50)(14,26,51)(15,27,52)(16,19,53)(17,20,54)(18,21,46), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,49)(2,48)(3,47)(4,46)(5,54)(6,53)(7,52)(8,51)(9,50)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,31)(20,30)(21,29)(22,28)(23,36)(24,35)(25,34)(26,33)(27,32) );

G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,46),(7,47),(8,48),(9,49),(10,32),(11,33),(12,34),(13,35),(14,36),(15,28),(16,29),(17,30),(18,31),(19,44),(20,45),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43)], [(2,36,42),(3,43,28),(5,30,45),(6,37,31),(8,33,39),(9,40,34),(11,23,48),(12,49,24),(14,26,51),(15,52,27),(17,20,54),(18,46,21)], [(1,35,41),(2,36,42),(3,28,43),(4,29,44),(5,30,45),(6,31,37),(7,32,38),(8,33,39),(9,34,40),(10,22,47),(11,23,48),(12,24,49),(13,25,50),(14,26,51),(15,27,52),(16,19,53),(17,20,54),(18,21,46)], [(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,49),(2,48),(3,47),(4,46),(5,54),(6,53),(7,52),(8,51),(9,50),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,45),(18,44),(19,31),(20,30),(21,29),(22,28),(23,36),(24,35),(25,34),(26,33),(27,32)]])

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 2 2 2 2 3 3 6 6 2 2 2 2 3 3 6 6 27 27 27 27 6 ··· 6 6 ··· 6

42 irreducible representations

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

Matrix representation of C2×C32⋊D9 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
,
 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 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0
,
 14 2 0 0 0 0 0 0 17 12 0 0 0 0 0 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18 0 0
,
 14 2 0 0 0 0 0 0 7 5 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 18 0 0 0 0 1 1 0 0 0 0 0 0 0 18 0 0 0 0 1 1 0 0 0 0 0 0 0 18 0 0 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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[14,17,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0],[14,7,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0] >;

C2×C32⋊D9 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes D_9
% in TeX

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

G:=SmallGroup(324,63);
// by ID

G=gap.SmallGroup(324,63);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,303,453,2164,7781]);
// 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,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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