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G = C5⋊D5  order 50 = 2·52

The semidirect product of C5 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, A-group

Aliases: C5⋊D5, C52⋊2C2, SmallGroup(50,4)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C5⋊D5
Chief series
C1 — C5 — C52 — C5⋊D5
Lower central
C52 — C5⋊D5
Upper central
C1

Generators and relations for C5⋊D5
 G = < a,b,c | a5=b5=c2=1, ab=ba, cac=a-1, cbc=b-1 >

C1
25C2
C5
C5
C5
C5
C5
C5
5D5
5D5
5D5
5D5
5D5
5D5
C52
C5⋊D5

Character table of C5⋊D5

 class 125A5B5C5D5E5F5G5H5I5J5K5L
 size 125222222222222
ρ111111111111111    trivial
ρ21-1111111111111    linear of order 2
ρ320-1+√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ420-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ520-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ620-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ720-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2    orthogonal lifted from D5
ρ8202-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/22    orthogonal lifted from D5
ρ920-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ1020-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2    orthogonal lifted from D5
ρ1120-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ1220-1-√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ1320-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ14202-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/22    orthogonal lifted from D5

Permutation representations of C5⋊D5
►On 25 points - transitive group 25T5
Generators in S25
(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)

(1 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)

(1 9)(2 8)(3 7)(4 6)(5 10)(11 22)(12 21)(13 25)(14 24)(15 23)(16 17)(18 20)
 
G:=sub<Sym(25)| (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), (1,24,19,14,9)(2,25,20,15,10)(3,21,16,11,6)(4,22,17,12,7)(5,23,18,13,8), (1,9)(2,8)(3,7)(4,6)(5,10)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20)>;
 
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), (1,24,19,14,9)(2,25,20,15,10)(3,21,16,11,6)(4,22,17,12,7)(5,23,18,13,8), (1,9)(2,8)(3,7)(4,6)(5,10)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20) );
 
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)], [(1,24,19,14,9),(2,25,20,15,10),(3,21,16,11,6),(4,22,17,12,7),(5,23,18,13,8)], [(1,9),(2,8),(3,7),(4,6),(5,10),(11,22),(12,21),(13,25),(14,24),(15,23),(16,17),(18,20)]])
 
G:=TransitiveGroup(25,5);
 

C5⋊D5 is a maximal subgroup of
 C5⋊F5  C52⋊C4  D52  C52⋊C6  C5⋊D15  C52⋊C10  C25⋊D5  C53⋊C2  C5⋊D35
C5⋊D5 is a maximal quotient of
 C52⋊6C4  C5⋊D15  C25⋊D5  He5⋊C2  C53⋊C2  C5⋊D35

Matrix representation of C5⋊D5 ►in GL4(𝔽11) generated by

8100
21000
0001
00103
,
101000
9800
0001
00103
,
0700
8000
00100
0081
G:=sub<GL(4,GF(11))| [8,2,0,0,1,10,0,0,0,0,0,10,0,0,1,3],[10,9,0,0,10,8,0,0,0,0,0,10,0,0,1,3],[0,8,0,0,7,0,0,0,0,0,10,8,0,0,0,1] >;
 

C5⋊D5 in GAP, Magma, Sage, TeX 

C_5\rtimes D_5
 
% in TeX
 
G:=Group("C5:D5");
 
// GroupNames label
 
G:=SmallGroup(50,4);
 
// by ID
 
G=gap.SmallGroup(50,4);
 
# by ID
 
G:=PCGroup([3,-2,-5,-5,49,362]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^5=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C5⋊D5 in TeX
Character table of C5⋊D5 in TeX

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