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## G = C5×He3order 135 = 33·5

### Direct product of C5 and He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C5×He3, C32⋊C15, C15.1C32, (C3×C15)⋊C3, C3.1(C3×C15), SmallGroup(135,3)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×He3
G = < a,b,c,d | a5=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Smallest permutation representation of C5×He3
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)
(6 42 13)(7 43 14)(8 44 15)(9 45 11)(10 41 12)(26 31 40)(27 32 36)(28 33 37)(29 34 38)(30 35 39)
(1 25 17)(2 21 18)(3 22 19)(4 23 20)(5 24 16)(6 13 42)(7 14 43)(8 15 44)(9 11 45)(10 12 41)(26 31 40)(27 32 36)(28 33 37)(29 34 38)(30 35 39)
(1 10 31)(2 6 32)(3 7 33)(4 8 34)(5 9 35)(11 39 24)(12 40 25)(13 36 21)(14 37 22)(15 38 23)(16 45 30)(17 41 26)(18 42 27)(19 43 28)(20 44 29)

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

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

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

C5×He3 is a maximal subgroup of   He3⋊D5  C32⋊D15

55 conjugacy classes

 class 1 3A 3B 3C ··· 3J 5A 5B 5C 5D 15A ··· 15H 15I ··· 15AN order 1 3 3 3 ··· 3 5 5 5 5 15 ··· 15 15 ··· 15 size 1 1 1 3 ··· 3 1 1 1 1 1 ··· 1 3 ··· 3

55 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C5 C15 He3 C5×He3 kernel C5×He3 C3×C15 He3 C32 C5 C1 # reps 1 8 4 32 2 8

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

 16 0 0 0 16 0 0 0 16
,
 1 0 0 0 5 0 0 0 25
,
 5 0 0 0 5 0 0 0 5
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(31))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,5,0,0,0,25],[5,0,0,0,5,0,0,0,5],[0,0,1,1,0,0,0,1,0] >;

C5×He3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(135,3);
// by ID

G=gap.SmallGroup(135,3);
# by ID

G:=PCGroup([4,-3,-3,-5,-3,385]);
// Polycyclic

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

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