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## G = He5⋊4C4order 500 = 22·53

### 4th semidirect product of He5 and C4 acting faithfully

Aliases: He54C4, C524F5, He5⋊C2.1C2, C5.2(C52⋊C4), Aut(5- 1+2), SmallGroup(500,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — He5 — He5⋊4C4
 Chief series C1 — C5 — C52 — He5 — He5⋊C2 — He5⋊4C4
 Lower central He5 — He5⋊4C4
 Upper central C1 — C5

Generators and relations for He54C4
G = < a,b,c,d | a5=b5=c5=d4=1, cac-1=ab=ba, dad-1=a2b-1c, bc=cb, bd=db, dcd-1=c3 >

25C2
5C5
5C5
10C5
10C5
25C4
5D5
5D5
10D5
10D5
25C10
2C52
2C52
5F5
5F5
25C20
10C5×D5
10C5×D5

Character table of He54C4

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 20G 20H size 1 25 25 25 1 1 1 1 20 20 20 20 20 20 25 25 25 25 25 25 25 25 25 25 25 25 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 i -i 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -i -i -i -i i i i i linear of order 4 ρ4 1 -1 -i i 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 4 ρ5 4 0 0 0 4 4 4 4 -1 -1 -1 -1 4 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F5 ρ6 4 0 0 0 4 4 4 4 -1 -1 -1 4 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F5 ρ7 4 0 0 0 4 4 4 4 3-√5/2 3+√5/2 -1-√5 -1 -1 -1+√5 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊C4 ρ8 4 0 0 0 4 4 4 4 -1-√5 -1+√5 3+√5/2 -1 -1 3-√5/2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊C4 ρ9 4 0 0 0 4 4 4 4 -1+√5 -1-√5 3-√5/2 -1 -1 3+√5/2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊C4 ρ10 4 0 0 0 4 4 4 4 3+√5/2 3-√5/2 -1+√5 -1 -1 -1-√5 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊C4 ρ11 5 1 -1 -1 5ζ52 5ζ53 5ζ5 5ζ54 0 0 0 0 0 0 ζ5 ζ52 ζ53 ζ54 -ζ5 -ζ54 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ52 complex faithful ρ12 5 1 1 1 5ζ52 5ζ53 5ζ5 5ζ54 0 0 0 0 0 0 ζ5 ζ52 ζ53 ζ54 ζ5 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 complex faithful ρ13 5 1 1 1 5ζ54 5ζ5 5ζ52 5ζ53 0 0 0 0 0 0 ζ52 ζ54 ζ5 ζ53 ζ52 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 complex faithful ρ14 5 1 1 1 5ζ53 5ζ52 5ζ54 5ζ5 0 0 0 0 0 0 ζ54 ζ53 ζ52 ζ5 ζ54 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 complex faithful ρ15 5 1 -1 -1 5ζ5 5ζ54 5ζ53 5ζ52 0 0 0 0 0 0 ζ53 ζ5 ζ54 ζ52 -ζ53 -ζ52 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ5 complex faithful ρ16 5 1 1 1 5ζ5 5ζ54 5ζ53 5ζ52 0 0 0 0 0 0 ζ53 ζ5 ζ54 ζ52 ζ53 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 complex faithful ρ17 5 1 -1 -1 5ζ53 5ζ52 5ζ54 5ζ5 0 0 0 0 0 0 ζ54 ζ53 ζ52 ζ5 -ζ54 -ζ5 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ53 complex faithful ρ18 5 1 -1 -1 5ζ54 5ζ5 5ζ52 5ζ53 0 0 0 0 0 0 ζ52 ζ54 ζ5 ζ53 -ζ52 -ζ53 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ54 complex faithful ρ19 5 -1 -i i 5ζ54 5ζ5 5ζ52 5ζ53 0 0 0 0 0 0 -ζ52 -ζ54 -ζ5 -ζ53 ζ4ζ52 ζ4ζ53 ζ4ζ54 ζ4ζ5 ζ43ζ5 ζ43ζ52 ζ43ζ53 ζ43ζ54 complex faithful ρ20 5 -1 -i i 5ζ53 5ζ52 5ζ54 5ζ5 0 0 0 0 0 0 -ζ54 -ζ53 -ζ52 -ζ5 ζ4ζ54 ζ4ζ5 ζ4ζ53 ζ4ζ52 ζ43ζ52 ζ43ζ54 ζ43ζ5 ζ43ζ53 complex faithful ρ21 5 -1 i -i 5ζ53 5ζ52 5ζ54 5ζ5 0 0 0 0 0 0 -ζ54 -ζ53 -ζ52 -ζ5 ζ43ζ54 ζ43ζ5 ζ43ζ53 ζ43ζ52 ζ4ζ52 ζ4ζ54 ζ4ζ5 ζ4ζ53 complex faithful ρ22 5 -1 i -i 5ζ54 5ζ5 5ζ52 5ζ53 0 0 0 0 0 0 -ζ52 -ζ54 -ζ5 -ζ53 ζ43ζ52 ζ43ζ53 ζ43ζ54 ζ43ζ5 ζ4ζ5 ζ4ζ52 ζ4ζ53 ζ4ζ54 complex faithful ρ23 5 -1 -i i 5ζ52 5ζ53 5ζ5 5ζ54 0 0 0 0 0 0 -ζ5 -ζ52 -ζ53 -ζ54 ζ4ζ5 ζ4ζ54 ζ4ζ52 ζ4ζ53 ζ43ζ53 ζ43ζ5 ζ43ζ54 ζ43ζ52 complex faithful ρ24 5 -1 i -i 5ζ52 5ζ53 5ζ5 5ζ54 0 0 0 0 0 0 -ζ5 -ζ52 -ζ53 -ζ54 ζ43ζ5 ζ43ζ54 ζ43ζ52 ζ43ζ53 ζ4ζ53 ζ4ζ5 ζ4ζ54 ζ4ζ52 complex faithful ρ25 5 -1 i -i 5ζ5 5ζ54 5ζ53 5ζ52 0 0 0 0 0 0 -ζ53 -ζ5 -ζ54 -ζ52 ζ43ζ53 ζ43ζ52 ζ43ζ5 ζ43ζ54 ζ4ζ54 ζ4ζ53 ζ4ζ52 ζ4ζ5 complex faithful ρ26 5 -1 -i i 5ζ5 5ζ54 5ζ53 5ζ52 0 0 0 0 0 0 -ζ53 -ζ5 -ζ54 -ζ52 ζ4ζ53 ζ4ζ52 ζ4ζ5 ζ4ζ54 ζ43ζ54 ζ43ζ53 ζ43ζ52 ζ43ζ5 complex faithful

Permutation representations of He54C4
On 25 points - transitive group 25T35
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 13 21 18 9)(2 14 22 19 10)(3 15 23 20 6)(4 11 24 16 7)(5 12 25 17 8)
(2 10 19 22 14)(3 20 15 6 23)(4 24 7 11 16)(5 12 25 17 8)
(2 4 8 23)(3 19 16 25)(5 20 14 11)(6 22 24 12)(7 17 15 10)

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

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

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

G:=TransitiveGroup(25,35);

Matrix representation of He54C4 in GL5(𝔽41)

 0 1 0 0 0 0 0 16 0 0 0 0 0 10 0 0 0 0 0 37 18 0 0 0 0
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 16 0 0 0 0 0 10 0 0 0 0 0 37 0 0 0 0 0 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(5,GF(41))| [0,0,0,0,18,1,0,0,0,0,0,16,0,0,0,0,0,10,0,0,0,0,0,37,0],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,10,0,0,0,0,0,37,0,0,0,0,0,18],[1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0] >;

He54C4 in GAP, Magma, Sage, TeX

{\rm He}_5\rtimes_4C_4
% in TeX

G:=Group("He5:4C4");
// GroupNames label

G:=SmallGroup(500,25);
// by ID

G=gap.SmallGroup(500,25);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,10,182,127,803,808,613]);
// Polycyclic

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

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