#72260044

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ #104
Sylow	C₅² ⋊ #825
Sylow	C₃ ⋊ #11716609

Subgroups

	subgroup	G/H
center	C₄	#10597121
commutator	C₃×C₄×C₅²	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₅×D₁₅
Fitting, cosocle	C₂×C₃×C₄×C₅²	C₂²
socle	C₂×C₃×C₅²	C₂²⋊C₄
prime radical	C₅×#6817	C₂
hypocenter	C₃×C₅²	#104
LCS	G ▷ C₃×C₄×C₅² ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

236 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 60 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,b^2=c,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^2,f^b=f^2,g^a=g^4,g^b=g^4,h^a=g^3h,h^b=h^4

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260052, #72260134, D₃×#11716609
maximal normal subgroup of	#72260052, #72260134, D₃×#11716609

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