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## G = D5×3- 1+2order 270 = 2·33·5

### Direct product of D5 and 3- 1+2

Aliases: D5×3- 1+2, C453C6, (C9×D5)⋊C3, (C3×C15).C6, C92(C3×D5), C32.(C3×D5), C15.3(C3×C6), (C32×D5).C3, C5⋊(C2×3- 1+2), C3.3(C32×D5), (C3×D5).3C32, (C5×3- 1+2)⋊3C2, SmallGroup(270,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — D5×3- 1+2
 Chief series C1 — C5 — C15 — C45 — C5×3- 1+2 — D5×3- 1+2
 Lower central C5 — C15 — D5×3- 1+2
 Upper central C1 — C3 — 3- 1+2

Generators and relations for D5×3- 1+2
G = < a,b,c,d | a5=b2=c9=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of D5×3- 1+2
On 45 points
Generators in S45
(1 36 23 39 15)(2 28 24 40 16)(3 29 25 41 17)(4 30 26 42 18)(5 31 27 43 10)(6 32 19 44 11)(7 33 20 45 12)(8 34 21 37 13)(9 35 22 38 14)
(1 15)(2 16)(3 17)(4 18)(5 10)(6 11)(7 12)(8 13)(9 14)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 22 25)(21 27 24)(28 34 31)(29 32 35)(37 43 40)(38 41 44)

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

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

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

44 conjugacy classes

 class 1 2 3A 3B 3C 3D 5A 5B 6A 6B 6C 6D 9A ··· 9F 15A 15B 15C 15D 15E 15F 15G 15H 18A ··· 18F 45A ··· 45L order 1 2 3 3 3 3 5 5 6 6 6 6 9 ··· 9 15 15 15 15 15 15 15 15 18 ··· 18 45 ··· 45 size 1 5 1 1 3 3 2 2 5 5 15 15 3 ··· 3 2 2 2 2 6 6 6 6 15 ··· 15 6 ··· 6

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 type + + + image C1 C2 C3 C3 C6 C6 D5 C3×D5 C3×D5 3- 1+2 C2×3- 1+2 D5×3- 1+2 kernel D5×3- 1+2 C5×3- 1+2 C9×D5 C32×D5 C45 C3×C15 3- 1+2 C9 C32 D5 C5 C1 # reps 1 1 6 2 6 2 2 12 4 2 2 4

Matrix representation of D5×3- 1+2 in GL5(𝔽181)

 0 1 0 0 0 180 167 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 167 180 0 0 0 0 0 180 0 0 0 0 0 180 0 0 0 0 0 180
,
 48 0 0 0 0 0 48 0 0 0 0 0 1 110 180 0 0 137 48 0 0 0 47 31 132
,
 132 0 0 0 0 0 132 0 0 0 0 0 1 110 0 0 0 0 132 121 0 0 0 0 48

G:=sub<GL(5,GF(181))| [0,180,0,0,0,1,167,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,167,0,0,0,0,180,0,0,0,0,0,180,0,0,0,0,0,180,0,0,0,0,0,180],[48,0,0,0,0,0,48,0,0,0,0,0,1,137,47,0,0,110,48,31,0,0,180,0,132],[132,0,0,0,0,0,132,0,0,0,0,0,1,0,0,0,0,110,132,0,0,0,0,121,48] >;

D5×3- 1+2 in GAP, Magma, Sage, TeX

D_5\times 3_-^{1+2}
% in TeX

G:=Group("D5xES-(3,1)");
// GroupNames label

G:=SmallGroup(270,7);
// by ID

G=gap.SmallGroup(270,7);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-5,187,57,5404]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^9=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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