#72230978

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₅² ⋊ C₂³
Hall	C₃×C₅² ⋊ C₂²×D₄
Sylow	C₅² ⋊ C₂×D₃×D₄
Sylow	C₃ ⋊ D₄×D₅²

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅²
commutator, socle, cosocle	C₂×C₃×C₅²	C₂⁴
Fitting	C₃×C₄×C₅²	C₂³
hypercenter	C₄	D₃×D₅²
hypocenter	C₃×C₅²	C₂²×D₄
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₄

1052 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₄×D₅², #72259806, #72259904, …
maximal normal subgroup of	D₃×D₄×D₅², #72259806, #72259904, …

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