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## G = C2×C32.5He3order 486 = 2·35

### Direct product of C2 and C32.5He3

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

Aliases: C2×C32.5He3, C9⋊C9.2C6, (C3×C6).5He3, (C3×C18).3C32, C32.5(C2×He3), C3.He3.5C6, C6.10(He3⋊C3), (C2×C9⋊C9).2C3, (C3×C9).3(C3×C6), C3.10(C2×He3⋊C3), (C2×C3.He3).2C3, SmallGroup(486,89)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C32.5He3
 Chief series C1 — C3 — C32 — C3×C9 — C9⋊C9 — C32.5He3 — C2×C32.5He3
 Lower central C1 — C3 — C32 — C3×C9 — C2×C32.5He3
 Upper central C1 — C6 — C3×C6 — C3×C18 — C2×C32.5He3

Generators and relations for C2×C32.5He3
G = < a,b,c,d,e,f | a2=b3=c3=1, d3=b-1, e3=c-1, f3=c, 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 >

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

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

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

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

38 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 9A ··· 9H 9I ··· 9N 18A ··· 18H 18I ··· 18N order 1 2 3 3 3 3 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 1 3 3 1 1 3 3 9 ··· 9 27 ··· 27 9 ··· 9 27 ··· 27

38 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 9 9 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 He3⋊C3 C2×He3⋊C3 C32.5He3 C2×C32.5He3 kernel C2×C32.5He3 C32.5He3 C2×C9⋊C9 C2×C3.He3 C9⋊C9 C3.He3 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 6 2 6 2 2 6 6 2 2

Matrix representation of C2×C32.5He3 in GL12(𝔽19)

 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
,
 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 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 7 7 7 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 12 12 12 0 0 0 0 0 11
,
 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
,
 6 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 9 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 7 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 11 11 11 12 12 4 0 0 0 0 0 0 7 8 0 7 7 7 0 0 0 0 0 0 1 1 1 0 0 0 12 12 4 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 11 18 0 0 0 0 7 0 7
,
 11 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 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 7 7 7 18 18 6 0 0 0 0 0 0 1 0 0 8 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 1 1 0 0 0 12 12 4 0 0 0 18 11 11 0 0 0 18 0 7
,
 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 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 7 7 7 18 18 6 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 12 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 1 1 1 0 0 8 0 0 0

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],[11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,7,0,0,12,0,0,0,0,1,0,0,0,7,0,0,12,0,0,0,0,0,1,0,0,7,0,0,12,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11],[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],[6,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,11,7,1,0,11,0,0,0,0,11,0,0,11,8,1,0,18,0,0,0,0,0,7,0,11,0,1,0,0,0,0,0,0,0,0,0,12,7,0,0,0,0,0,0,0,0,0,11,12,7,0,0,0,0,0,0,0,0,0,0,4,7,0,0,0,0,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,4,0,7],[11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,7,1,0,1,18,0,0,0,1,0,0,0,7,0,0,1,11,0,0,0,0,1,0,0,7,0,0,1,11,0,0,0,0,0,0,0,18,8,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12,18,0,0,0,0,0,0,0,0,0,7,12,0,0,0,0,0,0,0,0,0,0,0,4,7],[0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,7,0,1,0,0,0,0,0,7,0,0,0,0,7,1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,1,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0] >;

C2×C32.5He3 in GAP, Magma, Sage, TeX

C_2\times C_3^2._5{\rm He}_3
% in TeX

G:=Group("C2xC3^2.5He3");
// GroupNames label

G:=SmallGroup(486,89);
// by ID

G=gap.SmallGroup(486,89);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,224,338,873,735,453,3250]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=1,d^3=b^-1,e^3=c^-1,f^3=c,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

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