#72231176

order	14400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

derived, Hall	C₃²×C₅² ⋊ C₂×C₄×C₈
Sylow	C₅² ⋊ C₄×#3710
Sylow	C₃² ⋊ C₂×#11717300

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72224613
commutator, hypocenter	C₃²×C₅²	C₂×C₄×C₈
Fitting, socle	C₂×C₃²×C₅²	C₂×C₄²
prime radical	C₂×C₃⋊D₃×C₅⋊D₅	C₂×C₄
cosocle	C₃²×#6036	C₂³
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₂

1674 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1674 conjugacy classes of subgroups — too many to draw. Only its 195 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:

2^6,3^2,5^2|a^2=d,c^2=e,d^2=f,g^a=g^2,g^b=g^2,h^a=g^2h,h^b=h^2,i^a=i^3,i^c=i^3,i^d=i^4,i^e=i^4,j^a=i^3j,j^c=j^3,j^d=i^2j,j^e=j^4

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231278
maximal normal subgroup of	#72231278

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^3);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);