#72230560

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₅² ⋊ C₂×C₄
Hall	C₃×C₅² ⋊ C₄×D₄
Sylow	C₅² ⋊ #893
Sylow	C₃ ⋊ #11717363

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#10597183
commutator, socle	C₂×C₃×C₅²	C₂²×C₄
Fitting	C₃×C₄×C₅²	C₂×C₄
prime radical	#11925655	C₂
cosocle	C₂×C₃×C₅⋊D₅	C₂³
hypercenter	C₄	#10597183
hypocenter	C₃×C₅²	C₄×D₄
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₄

436 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 67 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=d,b^2=d,c^2=e,c^a=ce,f^b=f^2,g^a=g^2,g^b=g^3,g^d=g^4,h^a=g^3h^3,h^b=h^3,h^d=h^4

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11717363, #72230582, #72230650, …
maximal normal subgroup of	D₃×#11717363, #72230582, #72230650, …

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