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G = C2×C32.He3  order 486 = 2·35

Direct product of C2 and C32.He3

direct product, metabelian, nilpotent (class 4), monomial, 3-elementary

Aliases: C2×C32.He3, C9⋊C9⋊4C6, (C3×C6).4He3, He3⋊C3⋊6C6, C3.He3⋊3C6, (C3×C18).2C32, C32.4(C2×He3), C6.9(He3⋊C3), (C2×C9⋊C9)⋊1C3, (C3×C9).2(C3×C6), (C2×He3⋊C3)⋊2C3, C3.9(C2×He3⋊C3), (C2×C3.He3)⋊2C3, SmallGroup(486,88)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — C2×C32.He3
Chief series
C1 — C3 — C32 — C3×C9 — C9⋊C9 — C32.He3 — C2×C32.He3
Lower central
C1 — C3 — C32 — C3×C9 — C2×C32.He3
Upper central
C1 — C6 — C3×C6 — C3×C18 — C2×C32.He3

Generators and relations for C2×C32.He3
 G = < a,b,c,d,e,f | a2=b3=c3=f3=1, d3=b-1, e3=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, ede-1=cd=dc, ce=ec, cf=fc, fdf-1=bde2, fef-1=b-1ce >

C1
C2
C3
3C3
27C3
27C3
C6
3C6
27C6
27C6
C32
3C9
9C9
9C32
9C32
9C9
C3×C6
3C18
9C3×C6
9C18
9C3×C6
9C18
C3×C9
3He3
3C3×C9
33- 1+2
3He3
C3×C18
3C2×He3
3C3×C18
3C2×3- 1+2
3C2×He3
C9⋊C9
C3.He3
He3⋊C3
He3⋊C3
C2×C9⋊C9
C2×C3.He3
C2×He3⋊C3
C2×He3⋊C3
C32.He3
C2×C32.He3

Smallest permutation representation of C2×C32.He3
►On 54 points
Generators in S54
(1 4)(2 5)(3 6)(7 15)(8 13)(9 14)(10 16)(11 17)(12 18)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)

(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)

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

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

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

38 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H6A6B6C6D6E6F6G6H9A···9H9I9J18A···18H18I18J
order1233333333666666669···99918···181818
size111133272727271133272727279···927279···92727

38 irreducible representations

dim11111111333399
type++
imageC1C2C3C3C3C6C6C6He3C2×He3He3⋊C3C2×He3⋊C3C32.He3C2×C32.He3
kernelC2×C32.He3C32.He3C2×C9⋊C9C2×He3⋊C3C2×C3.He3C9⋊C9He3⋊C3C3.He3C3×C6C32C6C3C2C1
# reps11242242226622

Matrix representation of C2×C32.He3 ►in GL12(𝔽19)

1800000000000
0180000000000
0018000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
700000000000
070000000000
007000000000
000100000000
000010000000
000001000000
0000001100000
0000000110000
0000000011000
000000000700
000000000070
000000000007
,
100000000000
010000000000
001000000000
0001100000000
0000110000000
0000011000000
0000001100000
0000000110000
0000000011000
0000000001100
0000000000110
0000000000011
,
1700000000000
0170000000000
005000000000
000100000000
000070000000
00012711000000
000000070000
000000181215000
0000001117000
000000000181215
000000000700
000000000701
,
700000000000
010000000000
0011000000000
000010000000
00012810000000
00012011000000
000000010000
00000012810000
00000012011000
0000000000110
000000000181215
0000000001807
,
0011000000000
1100000000000
0110000000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
000100000000
000010000000
000001000000

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],[7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11],[17,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,18,11,0,0,0,0,0,0,0,0,0,7,12,1,0,0,0,0,0,0,0,0,0,0,15,7,0,0,0,0,0,0,0,0,0,0,0,0,18,7,7,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,15,0,1],[7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,0,0,0,15,7],[0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C2×C32.He3 in GAP, Magma, Sage, TeX 

C_2\times C_3^2.{\rm He}_3
 
% in TeX
 
G:=Group("C2xC3^2.He3");
 
// GroupNames label
 
G:=SmallGroup(486,88);
 
// by ID
 
G=gap.SmallGroup(486,88);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,338,873,735,453,3250]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=f^3=1,d^3=b^-1,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,b*d=d*b,b*e=e*b,f*b*f^-1=b*c^-1,e*d*e^-1=c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=b*d*e^2,f*e*f^-1=b^-1*c*e>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C32.He3 in TeX

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