#72213951

order	25000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#15691114 ⋊ C₂×C₄
derived	C₂₅² ⋊ C₂×C₄×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₂₅²	C₂×C₄×C₅
Frattini, socle	C₅²	C₅×D₅⋊AGL(1,𝔽₅)
Fitting	#15691114	C₂×C₄
prime radical	#72213905	C₂
cosocle	C₂₅⋊D₂₅	C₂²×C₅
LCS	G ▷ C₂₅²
UCS	C₁

270 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228896
maximal normal subgroup of	#72228896
holomorph of	D₂₅

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 5, 5, 5, 5]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 5, 2, F.5^3*F.7^3);;
gap> SetConjugate(coll, 5, 3, F.5*F.7^2);;
gap> SetConjugate(coll, 5, 4, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6^3*F.8^3);;
gap> SetConjugate(coll, 6, 3, F.6*F.8^2);;
gap> SetConjugate(coll, 6, 4, F.6^4*F.8^4);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);