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## G = C5×He3⋊C2order 270 = 2·33·5

### Direct product of C5 and He3⋊C2

Aliases: C5×He3⋊C2, He32C10, (C3×C15)⋊4S3, (C5×He3)⋊5C2, C322(C5×S3), C15.4(C3⋊S3), C3.2(C5×C3⋊S3), SmallGroup(270,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C5×He3⋊C2
 Chief series C1 — C3 — C32 — He3 — C5×He3 — C5×He3⋊C2
 Lower central He3 — C5×He3⋊C2
 Upper central C1 — C15

Generators and relations for C5×He3⋊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 >

Smallest permutation representation of C5×He3⋊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 3A 3B 3C 3D 3E 3F 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 15A ··· 15H 15I ··· 15X 30A ··· 30H order 1 2 3 3 3 3 3 3 5 5 5 5 6 6 10 10 10 10 15 ··· 15 15 ··· 15 30 ··· 30 size 1 9 1 1 6 6 6 6 1 1 1 1 9 9 9 9 9 9 1 ··· 1 6 ··· 6 9 ··· 9

50 irreducible representations

 dim 1 1 1 1 2 2 3 3 type + + + image C1 C2 C5 C10 S3 C5×S3 He3⋊C2 C5×He3⋊C2 kernel C5×He3⋊C2 C5×He3 He3⋊C2 He3 C3×C15 C32 C5 C1 # reps 1 1 4 4 4 16 4 16

Matrix representation of C5×He3⋊C2 in GL3(𝔽31) generated by

 8 0 0 0 8 0 0 0 8
,
 0 1 0 0 0 1 1 0 0
,
 5 0 0 0 5 0 0 0 5
,
 0 5 0 0 0 1 25 0 0
,
 1 0 0 0 0 1 0 1 0
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] >;

C5×He3⋊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

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