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G = He54C4order 500 = 22·53

4th semidirect product of He5 and C4 acting faithfully

non-abelian, supersoluble, monomial

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

Series: Derived Chief Lower central Upper central

C1C5He5 — He54C4
C1C5C52He5He5⋊C2 — He54C4
He5 — He54C4
C1C5

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
5C5×D5
5C5×D5
10C5×D5
10C5×D5
5C5×F5
5C5×F5

Character table of He54C4

 class 124A4B5A5B5C5D5E5F5G5H5I5J10A10B10C10D20A20B20C20D20E20F20G20H
 size 12525251111202020202020252525252525252525252525
ρ111111111111111111111111111    trivial
ρ211-1-111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31-1i-i1111111111-1-1-1-1-i-i-i-iiiii    linear of order 4
ρ41-1-ii1111111111-1-1-1-1iiii-i-i-i-i    linear of order 4
ρ540004444-1-1-1-14-1000000000000    orthogonal lifted from F5
ρ640004444-1-1-14-1-1000000000000    orthogonal lifted from F5
ρ7400044443-5/23+5/2-1-5-1-1-1+5000000000000    orthogonal lifted from C52⋊C4
ρ840004444-1-5-1+53+5/2-1-13-5/2000000000000    orthogonal lifted from C52⋊C4
ρ940004444-1+5-1-53-5/2-1-13+5/2000000000000    orthogonal lifted from C52⋊C4
ρ10400044443+5/23-5/2-1+5-1-1-1-5000000000000    orthogonal lifted from C52⋊C4
ρ1151-1-15253554000000ζ5ζ52ζ53ζ5455452535355452    complex faithful
ρ1251115253554000000ζ5ζ52ζ53ζ54ζ5ζ54ζ52ζ53ζ53ζ5ζ54ζ52    complex faithful
ρ1351115455253000000ζ52ζ54ζ5ζ53ζ52ζ53ζ54ζ5ζ5ζ52ζ53ζ54    complex faithful
ρ1451115352545000000ζ54ζ53ζ52ζ5ζ54ζ5ζ53ζ52ζ52ζ54ζ5ζ53    complex faithful
ρ1551-1-15545352000000ζ53ζ5ζ54ζ5253525545453525    complex faithful
ρ1651115545352000000ζ53ζ5ζ54ζ52ζ53ζ52ζ5ζ54ζ54ζ53ζ52ζ5    complex faithful
ρ1751-1-15352545000000ζ54ζ53ζ52ζ554553525254553    complex faithful
ρ1851-1-15455253000000ζ52ζ54ζ5ζ5352535455525354    complex faithful
ρ195-1-ii54552530000005254553ζ4ζ52ζ4ζ53ζ4ζ54ζ4ζ5ζ43ζ5ζ43ζ52ζ43ζ53ζ43ζ54    complex faithful
ρ205-1-ii53525450000005453525ζ4ζ54ζ4ζ5ζ4ζ53ζ4ζ52ζ43ζ52ζ43ζ54ζ43ζ5ζ43ζ53    complex faithful
ρ215-1i-i53525450000005453525ζ43ζ54ζ43ζ5ζ43ζ53ζ43ζ52ζ4ζ52ζ4ζ54ζ4ζ5ζ4ζ53    complex faithful
ρ225-1i-i54552530000005254553ζ43ζ52ζ43ζ53ζ43ζ54ζ43ζ5ζ4ζ5ζ4ζ52ζ4ζ53ζ4ζ54    complex faithful
ρ235-1-ii52535540000005525354ζ4ζ5ζ4ζ54ζ4ζ52ζ4ζ53ζ43ζ53ζ43ζ5ζ43ζ54ζ43ζ52    complex faithful
ρ245-1i-i52535540000005525354ζ43ζ5ζ43ζ54ζ43ζ52ζ43ζ53ζ4ζ53ζ4ζ5ζ4ζ54ζ4ζ52    complex faithful
ρ255-1i-i55453520000005355452ζ43ζ53ζ43ζ52ζ43ζ5ζ43ζ54ζ4ζ54ζ4ζ53ζ4ζ52ζ4ζ5    complex faithful
ρ265-1-ii55453520000005355452ζ4ζ53ζ4ζ52ζ4ζ5ζ4ζ54ζ43ζ54ζ43ζ53ζ43ζ52ζ43ζ5    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)

01000
001600
000100
000037
180000
,
160000
016000
001600
000160
000016
,
10000
016000
001000
000370
000018
,
10000
00100
00001
01000
00010

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

Export

Subgroup lattice of He54C4 in TeX
Character table of He54C4 in TeX

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