#72209683

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

Aliases

No aliases for this group.

Splittings

derived, Sylow

#10597831 ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#10597831⋊C₂
commutator, hypocenter	#10597831	C₄
Frattini	C₂×C₅²	C₅⋊D₅
Fitting, cosocle	C₂×#10597831	C₂
socle	C₂×C₅	#6953
prime radical	C₂×C₅³	D₅
LCS	G ▷ #10597831
UCS	C₁ ◁ C₂

81 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

SL(2,ℤ/25)

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