#72230998

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄×C₅

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ #127
Sylow	C₅² ⋊ #940
Sylow	C₃ ⋊ #11717260

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₃×C₅²	C₂³
Fitting	C₂×C₃×C₄×C₅²	C₂²
hypercenter	C₂×C₄	D₃×C₅⋊D₅
hypocenter	C₃×C₅²	#127
LCS	G ▷ C₂²×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

864 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 1200

Polycyclic presentation:

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#940, D₃×#11717260, #72231134
maximal normal subgroup of	C₅⋊D₅×#940, D₃×#11717260, #72231134

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