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G = C2×C32⋊D9  order 324 = 22·34

Direct product of C2 and C32⋊D9

direct product, metabelian, supersoluble, monomial

Aliases: C2×C32⋊D9, C32⋊2D18, C33.3D6, C9⋊S3⋊2C6, (C3×C6)⋊1D9, (C3×C18)⋊1C6, C6.4(C3×D9), C3.1(C6×D9), C6.2(C9⋊C6), C32⋊C9⋊3C22, (C32×C6).5S3, C6.5(C32⋊C6), C32.15(S3×C6), C3.(C2×C9⋊C6), (C2×C9⋊S3)⋊1C3, (C3×C9)⋊2(C2×C6), (C2×C32⋊C9)⋊2C2, (C3×C6).31(C3×S3), C3.1(C2×C32⋊C6), SmallGroup(324,63)

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 2A2B2C3A3B3C3D3E3F3G3H6A6B6C6D6E6F6G6H6I6J6K6L9A···9I18A···18I
order1222333333336666666666669···918···18
size1127272222336622223366272727276···66···6

42 irreducible representations

dim111111222222226666
type+++++++++++
imageC1C2C2C3C6C6S3D6D9C3×S3D18S3×C6C3×D9C6×D9C32⋊C6C9⋊C6C2×C32⋊C6C2×C9⋊C6
kernelC2×C32⋊D9C32⋊D9C2×C32⋊C9C2×C9⋊S3C9⋊S3C3×C18C32×C6C33C3×C6C3×C6C32C32C6C3C6C6C3C3
# reps121242113232661212

Matrix representation of C2×C32⋊D9 ►in GL8(𝔽19)

10000000
01000000
001800000
000180000
000018000
000001800
000000180
000000018
,
10000000
01000000
00100000
00010000
0000181800
00001000
00000001
0000001818
,
10000000
01000000
0018180000
00100000
0000181800
00001000
0000001818
00000010
,
142000000
1712000000
0000001818
00000010
00010000
0018180000
00000100
0000181800
,
142000000
75000000
00000011
000000018
00001100
000001800
00110000
000180000

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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