#72229811

order	12500
automorphism	?
homology	Abelianization (H₁) = C₄×C₅² · Schur multiplier (H₂) = C₅

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	5₋¹⁺⁴ ⋊ C₄
derived	C₅×C₂₅ ⋊ C₄×C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅×C₂₅	C₄×C₅²
Frattini, socle	C₅	C₅²×#1043
Fitting	5₋¹⁺⁴	C₄
prime radical	#11929399⋊C₅	C₂
cosocle	C₅⋊D₂₅	C₂×C₅²
LCS	G ▷ C₅×C₂₅
UCS	C₁

493 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 125

Polycyclic presentation:

2,5^2,2,5^3|a^2=d,f^5=g,b^a=bg^3,d^b=dg,e^a=e^2,e^b=eg^4,e^d=e^4,f^a=f^2g,f^b=fg^4,f^c=fg^4,f^d=f^4g^4,g^a=g^2,g^d=g^4

Conjugacy classes & character table

251 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72229822

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", "g");;
gap> coll := SingleCollector(F, [2, 5, 5, 2, 5, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7^3);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7^4);;
gap> SetConjugate(coll, 5, 4, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7^4);;
gap> SetConjugate(coll, 6, 3, F.6*F.7^4);;
gap> SetConjugate(coll, 6, 4, F.6^4*F.7^4);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);