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G = C5xHe3:C2order 270 = 2·33·5

Direct product of C5 and He3:C2

direct product, non-abelian, supersoluble, monomial

Aliases: C5xHe3:C2, He3:2C10, (C3xC15):4S3, (C5xHe3):5C2, C32:2(C5xS3), C15.4(C3:S3), C3.2(C5xC3:S3), SmallGroup(270,17)

Series: Derived Chief Lower central Upper central

C1C3He3 — C5xHe3:C2
C1C3C32He3C5xHe3 — C5xHe3:C2
He3 — C5xHe3:C2
C1C15

Generators and relations for C5xHe3:C2
 G = < a,b,c,d,e | a5=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 124 in 44 conjugacy classes, 16 normal (8 characteristic)
Quotients: C1, C2, C5, S3, C10, C3:S3, C5xS3, He3:C2, C5xC3:S3, C5xHe3:C2
9C2
3C3
3C3
3C3
3C3
3S3
3S3
3S3
3S3
9C6
9C10
3C15
3C15
3C15
3C15
3C3xS3
3C3xS3
3C3xS3
3C3xS3
3C5xS3
3C5xS3
3C5xS3
3C5xS3
9C30
3S3xC15
3S3xC15
3S3xC15
3S3xC15

Smallest permutation representation of C5xHe3:C2
On 45 points
Generators in S45
(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)
(1 42 18)(2 43 19)(3 44 20)(4 45 16)(5 41 17)(6 39 12)(7 40 13)(8 36 14)(9 37 15)(10 38 11)(21 33 26)(22 34 27)(23 35 28)(24 31 29)(25 32 30)
(1 8 33)(2 9 34)(3 10 35)(4 6 31)(5 7 32)(11 23 20)(12 24 16)(13 25 17)(14 21 18)(15 22 19)(26 42 36)(27 43 37)(28 44 38)(29 45 39)(30 41 40)
(1 26 21)(2 27 22)(3 28 23)(4 29 24)(5 30 25)(6 45 16)(7 41 17)(8 42 18)(9 43 19)(10 44 20)(11 35 38)(12 31 39)(13 32 40)(14 33 36)(15 34 37)
(11 38)(12 39)(13 40)(14 36)(15 37)(16 45)(17 41)(18 42)(19 43)(20 44)(21 26)(22 27)(23 28)(24 29)(25 30)

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

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

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

50 conjugacy classes

class 1  2 3A3B3C3D3E3F5A5B5C5D6A6B10A10B10C10D15A···15H15I···15X30A···30H
order123333335555661010101015···1515···1530···30
size1911666611119999991···16···69···9

50 irreducible representations

dim11112233
type+++
imageC1C2C5C10S3C5xS3He3:C2C5xHe3:C2
kernelC5xHe3:C2C5xHe3He3:C2He3C3xC15C32C5C1
# reps1144416416

Matrix representation of C5xHe3:C2 in GL3(F31) generated by

800
080
008
,
010
001
100
,
500
050
005
,
050
001
2500
,
100
001
010
G:=sub<GL(3,GF(31))| [8,0,0,0,8,0,0,0,8],[0,0,1,1,0,0,0,1,0],[5,0,0,0,5,0,0,0,5],[0,0,25,5,0,0,0,1,0],[1,0,0,0,0,1,0,1,0] >;

C5xHe3:C2 in GAP, Magma, Sage, TeX

C_5\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C5xHe3:C2");
// GroupNames label

G:=SmallGroup(270,17);
// by ID

G=gap.SmallGroup(270,17);
# by ID

G:=PCGroup([5,-2,-5,-3,-3,-3,302,1203,253]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Subgroup lattice of C5xHe3:C2 in TeX

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