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G = C52⋊C10order 250 = 2·53

The semidirect product of C52 and C10 acting faithfully

metabelian, supersoluble, monomial

Aliases: C52⋊C10, He51C2, C521D5, C5⋊D5⋊C5, C5.2(C5×D5), SmallGroup(250,5)

Series: Derived Chief Lower central Upper central

C1C52 — C52⋊C10
C1C5C52He5 — C52⋊C10
C52 — C52⋊C10
C1

Generators and relations for C52⋊C10
 G = < a,b,c | a5=b5=c10=1, ab=ba, cac-1=a-1b3, cbc-1=b-1 >

25C2
5C5
5C5
10C5
10C5
5D5
25C10
25D5
2C52
2C52
5C5×D5

Character table of C52⋊C10

 class 125A5B5C5D5E5F5G5H5I5J5K5L5M5N5O5P10A10B10C10D
 size 1252255551010101010101010101025252525
ρ11111111111111111111111    trivial
ρ21-11111111111111111-1-1-1-1    linear of order 2
ρ31-111ζ5ζ52ζ53ζ541ζ54ζ541ζ5ζ5ζ52ζ53ζ53ζ525254553    linear of order 10
ρ41111ζ5ζ52ζ53ζ541ζ54ζ541ζ5ζ5ζ52ζ53ζ53ζ52ζ52ζ54ζ5ζ53    linear of order 5
ρ51111ζ54ζ53ζ52ζ51ζ5ζ51ζ54ζ54ζ53ζ52ζ52ζ53ζ53ζ5ζ54ζ52    linear of order 5
ρ61-111ζ52ζ54ζ5ζ531ζ53ζ531ζ52ζ52ζ54ζ5ζ5ζ545453525    linear of order 10
ρ71-111ζ54ζ53ζ52ζ51ζ5ζ51ζ54ζ54ζ53ζ52ζ52ζ535355452    linear of order 10
ρ81111ζ52ζ54ζ5ζ531ζ53ζ531ζ52ζ52ζ54ζ5ζ5ζ54ζ54ζ53ζ52ζ5    linear of order 5
ρ91111ζ53ζ5ζ54ζ521ζ52ζ521ζ53ζ53ζ5ζ54ζ54ζ5ζ5ζ52ζ53ζ54    linear of order 5
ρ101-111ζ53ζ5ζ54ζ521ζ52ζ521ζ53ζ53ζ5ζ54ζ54ζ55525354    linear of order 10
ρ1120222222-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/20000    orthogonal lifted from D5
ρ1220222222-1+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+5/20000    orthogonal lifted from D5
ρ1320225254553-1+5/2ζ5+1ζ5452-1-5/2ζ54+1ζ535ζ525ζ5453ζ52+1ζ53+10000    complex lifted from C5×D5
ρ1420225254553-1-5/2ζ5452ζ5+1-1+5/2ζ535ζ54+1ζ53+1ζ52+1ζ5453ζ5250000    complex lifted from C5×D5
ρ1520225525354-1-5/2ζ53+1ζ525-1+5/2ζ52+1ζ5453ζ535ζ5452ζ5+1ζ54+10000    complex lifted from C5×D5
ρ1620225453525-1+5/2ζ5453ζ52+1-1-5/2ζ525ζ53+1ζ5+1ζ54+1ζ535ζ54520000    complex lifted from C5×D5
ρ1720225355452-1+5/2ζ54+1ζ535-1-5/2ζ5+1ζ5452ζ5453ζ525ζ53+1ζ52+10000    complex lifted from C5×D5
ρ1820225525354-1+5/2ζ525ζ53+1-1-5/2ζ5453ζ52+1ζ54+1ζ5+1ζ5452ζ5350000    complex lifted from C5×D5
ρ1920225355452-1-5/2ζ535ζ54+1-1+5/2ζ5452ζ5+1ζ52+1ζ53+1ζ525ζ54530000    complex lifted from C5×D5
ρ2020225453525-1-5/2ζ52+1ζ5453-1+5/2ζ53+1ζ525ζ5452ζ535ζ54+1ζ5+10000    complex lifted from C5×D5
ρ21100-5+55/2-5-55/2000000000000000000    orthogonal faithful
ρ22100-5-55/2-5+55/2000000000000000000    orthogonal faithful

Permutation representations of C52⋊C10
On 25 points - transitive group 25T23
Generators in S25
(1 10 16 21 15)(2 14 18 19 11)(3 6 24 23 9)(4 8 20 17 7)(5 12 22 25 13)
(1 5 2 3 4)(6 8 10 12 14)(7 15 13 11 9)(16 22 18 24 20)(17 21 25 19 23)
(2 3)(4 5)(6 7 8 9 10 11 12 13 14 15)(16 17 18 19 20 21 22 23 24 25)

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

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

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

G:=TransitiveGroup(25,23);

On 25 points - transitive group 25T24
Generators in S25
(1 23 8 13 18)(2 24 9 14 19)(3 10 20 25 15)(5 7 17 22 12)
(1 18 13 8 23)(2 24 9 14 19)(3 20 15 10 25)(4 16 11 6 21)(5 22 7 12 17)
(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)

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

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

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

G:=TransitiveGroup(25,24);

C52⋊C10 is a maximal subgroup of   C52⋊C20  He5⋊C4  C52⋊D10
C52⋊C10 is a maximal quotient of   He55C4

Matrix representation of C52⋊C10 in GL10(𝔽11)

0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
1000000000
0100000000
,
0100000000
10300000000
0001000000
00103000000
0000010000
00001030000
0000000100
00000010300
0000000001
00000000103
,
1000000000
31000000000
0000000088
00000000103
0000000100
0000001000
00003100000
0000880000
00103000000
0001000000

G:=sub<GL(10,GF(11))| [0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3],[1,3,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,3,8,0,0,0,0,0,0,0,0,10,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,10,0,0,0,0,0,0,0,0,8,3,0,0,0,0,0,0] >;

C52⋊C10 in GAP, Magma, Sage, TeX

C_5^2\rtimes C_{10}
% in TeX

G:=Group("C5^2:C10");
// GroupNames label

G:=SmallGroup(250,5);
// by ID

G=gap.SmallGroup(250,5);
# by ID

G:=PCGroup([4,-2,-5,-5,-5,482,366,3203]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^10=1,a*b=b*a,c*a*c^-1=a^-1*b^3,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C52⋊C10 in TeX
Character table of C52⋊C10 in TeX

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