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G = C536C4order 500 = 22·53

6th semidirect product of C53 and C4 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C536C4, C527F5, C526Dic5, C5⋊D5.2D5, C51(D5.D5), C52(C52⋊C4), (C5×C5⋊D5).3C2, SmallGroup(500,46)

Series: Derived Chief Lower central Upper central

C1C53 — C536C4
C1C5C52C53C5×C5⋊D5 — C536C4
C53 — C536C4
C1

Generators and relations for C536C4
 G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b2, dcd-1=c3 >

25C2
2C5
2C5
4C5
4C5
4C5
4C5
4C5
4C5
125C4
5D5
5D5
10D5
10D5
25C10
2C52
2C52
4C52
4C52
4C52
4C52
4C52
4C52
25F5
25Dic5
25F5
5C5×D5
5C5×D5
10C5×D5
10C5×D5
5C52⋊C4
5D5.D5
5D5.D5

Permutation representations of C536C4
On 20 points - transitive group 20T125
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 4 2 5 3)(6 8 10 7 9)(11 12 13 14 15)(16 20 19 18 17)
(1 3 5 2 4)(6 9 7 10 8)(11 12 13 14 15)(16 20 19 18 17)
(1 17 8 12)(2 16 9 11)(3 20 10 15)(4 19 6 14)(5 18 7 13)

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

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

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

G:=TransitiveGroup(20,125);

38 conjugacy classes

class 1  2 4A4B5A5B5C···5AF10A10B
order1244555···51010
size125125125224···45050

38 irreducible representations

dim111224444
type+++-++
imageC1C2C4D5Dic5F5D5.D5C52⋊C4C536C4
kernelC536C4C5×C5⋊D5C53C5⋊D5C52C52C5C5C1
# reps1122228416

Matrix representation of C536C4 in GL4(𝔽41) generated by

16000
01600
3331180
3133018
,
37000
01000
351180
2317016
,
10000
03700
913180
112016
,
53790
37509
00364
00436
G:=sub<GL(4,GF(41))| [16,0,33,31,0,16,31,33,0,0,18,0,0,0,0,18],[37,0,35,23,0,10,1,17,0,0,18,0,0,0,0,16],[10,0,9,11,0,37,13,2,0,0,18,0,0,0,0,16],[5,37,0,0,37,5,0,0,9,0,36,4,0,9,4,36] >;

C536C4 in GAP, Magma, Sage, TeX

C_5^3\rtimes_6C_4
% in TeX

G:=Group("C5^3:6C4");
// GroupNames label

G:=SmallGroup(500,46);
// by ID

G=gap.SmallGroup(500,46);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1203,808,5004,5009]);
// Polycyclic

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

Export

Subgroup lattice of C536C4 in TeX

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