#72209685

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₅)⋊C₅ ⋊ C₄

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₅	#92535
commutator, Fitting, hypocenter	H(3,𝔽₅)⋊C₅	C₄
Frattini	C₅²	#1042
prime radical, cosocle	#6953⋊C₅	C₂
LCS	G ▷ H(3,𝔽₅)⋊C₅
UCS	C₁ ◁ C₅

47 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72209687

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> SetConjugate(coll, 3, 1, F.3^4*F.5^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.5^4);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.5*F.6^2);;
gap> SetConjugate(coll, 4, 2, F.4^4*F.6^3);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5^3);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);