#72228836

order	2500
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

C₂₅² ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₂₅²	C₄
Frattini, socle	C₅²	#1042
prime radical	C₅×D₂₅	D₅
cosocle	C₂₅×D₂₅	C₂
LCS	G ▷ C₂₅²
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 50 points:

(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,18)(12,20)(14,22)(17,23)(19,24)(21,25)(27,31,32,41)(28,34,37,43)(29,30,39,40)(33,46,44,45)(35,38,42,47)(36,49,48,50);(1,6,21,20,4,3,14,24,10,9,8,23,18,11,17,16,15,5,19,22,7,2,12,25,13)(26,36,41,44,45,29,43,50,28,42,40,27,47,38,32,30,35,37,49,34,39,46,33,31,48)

Polycyclic presentation:

2^2,5^4|a^2=b,c^5=e,d^5=f,c^a=c^3e^3,c^b=c^4e^4,d^a=c^2d^4e^2f^4,d^b=c^4d,e^a=e^3,e^b=e^4,f^a=e^2f^4,f^b=e^4f

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228853, D₂₅×#1035
maximal normal subgroup of	#72228853, D₂₅×#1035

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 5, 5, 5, 5]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^3*F.5^3);;
gap> SetConjugate(coll, 3, 2, F.3^4*F.5^4);;
gap> SetConjugate(coll, 4, 1, F.3^2*F.4^4*F.5^2*F.6^4);;
gap> SetConjugate(coll, 4, 2, F.3^4*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^3);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6^4);;
gap> SetConjugate(coll, 6, 2, F.5^4*F.6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);