#72231278

order	28800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅² ⋊ C₂×C₄×M₁₆
Sylow	C₅² ⋊ C₄×#63818
Sylow	C₃² ⋊ C₂×#13107464

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#72224613
commutator, socle	C₂×C₃²×C₅²	C₂²×C₄²
Fitting	C₂²×C₃²×C₅²	C₂×C₄²
prime radical	C₂²×C₃⋊D₃×C₅⋊D₅	C₂×C₄
cosocle	C₃²×#6036	C₂⁴
hypercenter	C₂²	#72224613
hypocenter	C₃²×C₅²	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₃²×C₅² ▷ C₃²×C₅²
UCS	C₁ ◁ C₂ ◁ C₂²

6952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

2^7,3^2,5^2|a^2=e,d^2=f,e^2=g,b^a=bg,h^a=h^2,h^c=h^2,i^a=h^2i,i^c=i^2,j^a=j^3,j^d=j^3,j^e=j^4,j^f=j^4,k^a=j^3k,k^d=k^3,k^e=j^2k,k^f=k^4

Conjugacy classes & character table

312 conjugacy classes — view the full character table.

Relationships

holomorph of

C₃⋊#170

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> SetConjugate(coll, 10, 6, F.10^4);;
gap> SetConjugate(coll, 11, 1, F.10^3*F.11);;
gap> SetConjugate(coll, 11, 4, F.11^3);;
gap> SetConjugate(coll, 11, 5, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 6, F.11^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);