#72259604

order	10000
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₅⁴ ⋊ C₄²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅⁴	C₄²
prime radical, cosocle	C₅⋊D₅²	C₂²
LCS	G ▷ C₅⁴
UCS	C₁

968 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 200

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

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