#72259203

order	4000
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅³ ⋊ C₄×D₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×D₅⋊AGL(1,𝔽₅)
commutator, Fitting, socle	C₂×C₅³	C₂²×C₄
prime radical	#14522770	C₂
cosocle	C₂×C₅×C₅⋊D₅	C₂³
hypocenter	C₅³	C₄×D₄
LCS	G ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₂

536 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 80

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#11717375
maximal normal subgroup of	D₅×#11717375

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