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## G = D5×He3order 270 = 2·33·5

### Direct product of D5 and He3

Aliases: D5×He3, C5⋊(C2×He3), (C3×C15)⋊2C6, (C5×He3)⋊3C2, (C32×D5)⋊C3, C15.2(C3×C6), C322(C3×D5), C3.2(C32×D5), (C3×D5).2C32, SmallGroup(270,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — D5×He3
 Chief series C1 — C5 — C15 — C3×C15 — C5×He3 — D5×He3
 Lower central C5 — C15 — D5×He3
 Upper central C1 — C3 — He3

Generators and relations for D5×He3
G = < a,b,c,d,e | a5=b2=c3=d3=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Smallest permutation representation of D5×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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)
(1 34 19)(2 35 20)(3 31 16)(4 32 17)(5 33 18)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 30)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)

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

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

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

44 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 5A 5B 6A 6B 6C ··· 6J 15A 15B 15C 15D 15E ··· 15T order 1 2 3 3 3 ··· 3 5 5 6 6 6 ··· 6 15 15 15 15 15 ··· 15 size 1 5 1 1 3 ··· 3 2 2 5 5 15 ··· 15 2 2 2 2 6 ··· 6

44 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 type + + + image C1 C2 C3 C6 D5 C3×D5 He3 C2×He3 D5×He3 kernel D5×He3 C5×He3 C32×D5 C3×C15 He3 C32 D5 C5 C1 # reps 1 1 8 8 2 16 2 2 4

Matrix representation of D5×He3 in GL5(𝔽31)

 12 1 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 30 0 0 0 0 0 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 24 30 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5
,
 25 0 0 0 0 0 25 0 0 0 0 0 1 1 26 0 0 0 25 0 0 0 0 0 5

G:=sub<GL(5,GF(31))| [12,30,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,24,0,0,0,0,30,0,0,0,1,30],[1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[25,0,0,0,0,0,25,0,0,0,0,0,1,0,0,0,0,1,25,0,0,0,26,0,5] >;

D5×He3 in GAP, Magma, Sage, TeX

D_5\times {\rm He}_3
% in TeX

G:=Group("D5xHe3");
// GroupNames label

G:=SmallGroup(270,6);
// by ID

G=gap.SmallGroup(270,6);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-5,187,5404]);
// Polycyclic

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

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