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G = D45⋊C3  order 270 = 2·33·5

The semidirect product of D45 and C3 acting faithfully

metacyclic, supersoluble, monomial

Aliases: D45⋊C3, C45⋊1C6, C32.D15, 3- 1+2⋊D5, C5⋊(C9⋊C6), C9⋊(C3×D5), C15.3(C3×S3), (C3×C15).1S3, C3.3(C3×D15), (C5×3- 1+2)⋊1C2, SmallGroup(270,15)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C45 — D45⋊C3
Chief series
C1 — C3 — C15 — C45 — C5×3- 1+2 — D45⋊C3
Lower central
C45 — D45⋊C3
Upper central
C1

Generators and relations for D45⋊C3
 G = < a,b,c | a45=b2=c3=1, bab=a-1, cac-1=a31, bc=cb >

C1
45C2
C3
3C3
C5
15S3
45C6
C9
C32
2C9
9D5
C15
3C15
5D9
15C3×S3
3- 1+2
3D15
9C3×D5
C45
C3×C15
2C45
5C9⋊C6
D45
3C3×D15
C5×3- 1+2
D45⋊C3

Smallest permutation representation of D45⋊C3
►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)

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

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

32 conjugacy classes

class 1  2 3A3B3C5A5B6A6B9A9B9C15A15B15C15D15E15F15G15H45A···45L
order123335566999151515151515151545···45
size145233224545666222266666···6

32 irreducible representations

dim111122222266
type+++++++
imageC1C2C3C6S3D5C3×S3C3×D5D15C3×D15C9⋊C6D45⋊C3
kernelD45⋊C3C5×3- 1+2D45C45C3×C153- 1+2C15C9C32C3C5C1
# reps112212244814

Matrix representation of D45⋊C3 ►in GL8(𝔽181)

10265000000
11697000000
00000100
000018018000
00000001
000000180180
00100000
00010000
,
10000000
167180000000
00100000
001801800000
00000001
00000010
00000100
00001000
,
480000000
048000000
00100000
00010000
000018018000
00001000
00000001
000000180180

G:=sub<GL(8,GF(181))| [102,116,0,0,0,0,0,0,65,97,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,180,0,0,0,0,0,0,1,180,0,0,0,0,0,0,0,0,0,180,0,0,0,0,0,0,1,180,0,0],[1,167,0,0,0,0,0,0,0,180,0,0,0,0,0,0,0,0,1,180,0,0,0,0,0,0,0,180,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[48,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,180,1,0,0,0,0,0,0,180,0,0,0,0,0,0,0,0,0,0,180,0,0,0,0,0,0,1,180] >;
 

D45⋊C3 in GAP, Magma, Sage, TeX 

D_{45}\rtimes C_3
 
% in TeX
 
G:=Group("D45:C3");
 
// GroupNames label
 
G:=SmallGroup(270,15);
 
// by ID
 
G=gap.SmallGroup(270,15);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-5,-3,1532,727,462,1443,4504]);
 
// Polycyclic
 
G:=Group<a,b,c|a^45=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^31,b*c=c*b>;
 
// generators/relations
 

Export

Subgroup lattice of D45⋊C3 in TeX

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