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G = C2×C32.C33  order 486 = 2·35

Direct product of C2 and C32.C33

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

Aliases: C2×C32.C33, C6.15(C3×He3), C3.15(C6×He3), (C3×C6).12He3, (C3×C6).9C33, C3.He3⋊5C6, C33.17(C3×C6), (C3×C18).14C32, C32.12(C2×He3), C32.9(C32×C6), (C32×C6).16C32, (C6×3- 1+2).9C3, 3- 1+2.3(C3×C6), (C3×3- 1+2).12C6, (C2×3- 1+2).3C32, (C3×C9).8(C3×C6), (C2×C3.He3)⋊4C3, SmallGroup(486,218)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C32.C33
Chief series
C1 — C3 — C32 — C33 — C3×3- 1+2 — C32.C33 — C2×C32.C33
Lower central
C1 — C3 — C32 — C2×C32.C33
Upper central
C1 — C6 — C32×C6 — C2×C32.C33

Generators and relations for C2×C32.C33
 G = < a,b,c,d,e,f | a2=b3=c3=f3=1, d3=c, e3=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, ede-1=bc-1d, df=fd >

Subgroups: 252 in 124 conjugacy classes, 66 normal (12 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×C18, C3×C18, C2×3- 1+2, C2×3- 1+2, C32×C6, C3.He3, C3×3- 1+2, C3×3- 1+2, C2×C3.He3, C6×3- 1+2, C6×3- 1+2, C32.C33, C2×C32.C33
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C32.C33, C2×C32.C33

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

(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)(28 34 31)(29 32 35)(37 40 43)(39 45 42)(47 53 50)(48 51 54)

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

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

70 conjugacy classes

class 1  2 3A3B3C···3J6A6B6C···6J9A···9X18A···18X
order12333···3666···69···918···18
size11113···3113···39···99···9

70 irreducible representations

dim1111113399
type++
imageC1C2C3C3C6C6He3C2×He3C32.C33C2×C32.C33
kernelC2×C32.C33C32.C33C2×C3.He3C6×3- 1+2C3.He3C3×3- 1+2C3×C6C32C2C1
# reps111881886622

Matrix representation of C2×C32.C33 ►in GL9(𝔽19)

1800000000
0180000000
0018000000
0001800000
0000180000
0000018000
0000001800
0000000180
0000000018
,
100181818777
010000000
001000000
000700000
000070000
000007000
0000001100
0000000110
0000000011
,
700000000
070000000
007000000
000700000
000070000
000007000
000000700
000000070
000000007
,
1100777111
000010000
000001000
000000100
000000010
000000001
131212181818888
070000000
007000000
,
1118180881211
001000000
1588121212181818
000010000
000001000
0001100000
000000070
000000007
000000100
,
112101811087
070000000
0011000000
000100000
000070000
0000011000
000000100
000000070
0000000011

G:=sub<GL(9,GF(19))| [18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,7,0,0,0,0,0,18,0,0,0,7,0,0,0,0,18,0,0,0,0,7,0,0,0,7,0,0,0,0,0,11,0,0,7,0,0,0,0,0,0,11,0,7,0,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[11,0,0,0,0,0,13,0,0,0,0,0,0,0,0,12,7,0,0,0,0,0,0,0,12,0,7,7,0,0,0,0,0,18,0,0,7,1,0,0,0,0,18,0,0,7,0,1,0,0,0,18,0,0,1,0,0,1,0,0,8,0,0,1,0,0,0,1,0,8,0,0,1,0,0,0,0,1,8,0,0],[11,0,15,0,0,0,0,0,0,18,0,8,0,0,0,0,0,0,18,1,8,0,0,0,0,0,0,0,0,12,0,0,11,0,0,0,8,0,12,1,0,0,0,0,0,8,0,12,0,1,0,0,0,0,12,0,18,0,0,0,0,0,1,1,0,18,0,0,0,7,0,0,1,0,18,0,0,0,0,7,0],[1,0,0,0,0,0,0,0,0,12,7,0,0,0,0,0,0,0,1,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,7,0,0,0,0,11,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,7,0,7,0,0,0,0,0,0,0,11] >;
 

C2×C32.C33 in GAP, Magma, Sage, TeX 

C_2\times C_3^2.C_3^3
 
% in TeX
 
G:=Group("C2xC3^2.C3^3");
 
// GroupNames label
 
G:=SmallGroup(486,218);
 
// by ID
 
G=gap.SmallGroup(486,218);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,986,735,3250]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=f^3=1,d^3=c,e^3=c^-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,d*b*d^-1=b*c^-1,b*e=e*b,b*f=f*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,e*d*e^-1=b*c^-1*d,d*f=f*d>;
 
// generators/relations
 

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