#72259182

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

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₅⁴ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅⁴	C₄
prime radical, cosocle	C₅⋊(C₅⋊(C₅⋊D₅))	C₂
LCS	G ▷ C₅⁴
UCS	C₁

3360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 625

Polycyclic presentation:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259214, #72259487
maximal normal subgroup of	#72259214, #72259487

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