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G = D5×3- 1+2  order 270 = 2·33·5

Direct product of D5 and 3- 1+2

direct product, metacyclic, supersoluble, monomial

Aliases: D5×3- 1+2, C45⋊3C6, (C9×D5)⋊C3, (C3×C15).C6, C9⋊2(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 >

C1
5C2
C3
3C3
C5
5C6
15C6
C32
C9
C9
C9
D5
C15
3C15
5C18
5C18
5C3×C6
5C18
3- 1+2
C3×D5
3C3×D5
C45
C45
C45
C3×C15
5C2×3- 1+2
C9×D5
C9×D5
C32×D5
C9×D5
C5×3- 1+2
D5×3- 1+2

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 3A3B3C3D5A5B6A6B6C6D9A···9F15A15B15C15D15E15F15G15H18A···18F45A···45L
order1233335566669···9151515151515151518···1845···45
size151133225515153···32222666615···156···6

44 irreducible representations

dim111111222336
type+++
imageC1C2C3C3C6C6D5C3×D5C3×D53- 1+2C2×3- 1+2D5×3- 1+2
kernelD5×3- 1+2C5×3- 1+2C9×D5C32×D5C45C3×C153- 1+2C9C32D5C5C1
# reps1162622124224

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

01000
180167000
00100
00010
00001
,
10000
167180000
0018000
0001800
0000180
,
480000
048000
001110180
00137480
004731132
,
1320000
0132000
0011100
000132121
000048

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
 

Export

Subgroup lattice of D5×3- 1+2 in TeX

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