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G = C2×C33⋊C32  order 486 = 2·35

Direct product of C2 and C33⋊C32

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Aliases: C2×C33⋊C32, C3≀C3⋊4C6, (C6×He3)⋊5C3, C33⋊5(C3×C6), (C3×C6).9He3, (C3×He3)⋊17C6, (C3×C6).6C33, C3.12(C6×He3), C6.12(C3×He3), He3.10(C3×C6), (C32×C6)⋊2C32, C32.9(C2×He3), (C2×He3).3C32, C32.6(C32×C6), (C6×3- 1+2)⋊7C3, 3- 1+2⋊3(C3×C6), (C3×3- 1+2)⋊14C6, (C2×3- 1+2)⋊3C32, (C2×C3≀C3)⋊1C3, SmallGroup(486,215)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C33⋊C32
Chief series
C1 — C3 — C32 — C33 — C3×He3 — C33⋊C32 — C2×C33⋊C32
Lower central
C1 — C3 — C32 — C2×C33⋊C32
Upper central
C1 — C6 — C32×C6 — C2×C33⋊C32

Generators and relations for C2×C33⋊C32
 G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, ebe-1=bc-1d, cd=dc, ece-1=cd-1, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 468 in 148 conjugacy classes, 66 normal (16 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C33, C3×C18, C2×He3, C2×He3, C2×3- 1+2, C2×3- 1+2, C32×C6, C32×C6, C32×C6, C3≀C3, C3×He3, C3×3- 1+2, C2×C3≀C3, C6×He3, C6×3- 1+2, C33⋊C32, C2×C33⋊C32
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C33⋊C32, C2×C33⋊C32

Smallest permutation representation of C2×C33⋊C32
►On 54 points
Generators in S54
(1 32)(2 33)(3 31)(4 47)(5 48)(6 46)(7 34)(8 35)(9 36)(10 30)(11 28)(12 29)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 39)(20 37)(21 38)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)

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

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

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

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

70 conjugacy classes

class 1  2 3A3B3C···3J3K···3V6A6B6C···6J6K···6V9A···9L18A···18L
order12333···33···3666···66···69···918···18
size11113···39···9113···39···99···99···9

70 irreducible representations

dim111111113399
type++
imageC1C2C3C3C3C6C6C6He3C2×He3C33⋊C32C2×C33⋊C32
kernelC2×C33⋊C32C33⋊C32C2×C3≀C3C6×He3C6×3- 1+2C3≀C3C3×He3C3×3- 1+2C3×C6C32C2C1
# reps11184418446622

Matrix representation of C2×C33⋊C32 ►in GL12(𝔽19)

1800000000000
0180000000000
0018000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
070000000000
007000000000
700000000000
000000100000
0008101160000
000000071000
000000000100
0000000180010
000000070001
000100000000
0000000180000
000001070000
,
700000000000
070000000000
007000000000
000010000000
0008126000000
000007000000
0008101160000
0000018081000
0000070120000
0008100001160
0000018000081
0000070000120
,
100000000000
010000000000
001000000000
000700000000
000070000000
000007000000
000000700000
000000070000
000000007000
000000000700
000000000070
000000000007
,
1100000000000
070000000000
001000000000
000100000000
0000110000000
0008117000000
0007801104000
000812018018000
00001801118000
00012110000790
00001100008127
00080000011180
,
700000000000
070000000000
007000000000
000100000000
000010000000
000001000000
000000700000
000810070000
0001120007000
0000000001100
0007800000110
0008100000011

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

C2×C33⋊C32 in GAP, Magma, Sage, TeX 

C_2\times C_3^3\rtimes C_3^2
 
% in TeX
 
G:=Group("C2xC3^3:C3^2");
 
// GroupNames label
 
G:=SmallGroup(486,215);
 
// by ID
 
G=gap.SmallGroup(486,215);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,735,3250]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^3=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,e*b*e^-1=b*c^-1*d,c*d=d*c,e*c*e^-1=c*d^-1,c*f=f*c,d*e=e*d,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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