#72277867

order	2480
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₈×C₃₁ ⋊ C₂×C₅
Hall	#92470 ⋊ C₅
Hall	ΓL(1,𝔽₃₂) ⋊ SD₁₆
Sylow	C₃₁ ⋊ C₅×SD₁₆

Subgroups

	subgroup	G/H
center	C₂	C₄⋊(D₃₁⋊C₅)
commutator, cosocle	C₄×C₃₁	C₂²×C₅
Frattini	C₄	C₂×D₃₁⋊C₅
Fitting	C₈×C₃₁	C₂×C₅
socle	C₂×C₃₁	C₅×D₄
prime radical	C₄⋊(D₃₁⋊C₅)	C₂
hypercenter	C₈	D₃₁⋊C₅
hypocenter	C₃₁	C₅×SD₁₆
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 248

Polycyclic presentation:

2^2,5,2^2,31|a^2=d,d^2=e,b^a=bde,d^b=de,f^b=f^30,f^c=f^2

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277912
maximal normal subgroup of	#72277912

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");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 2, 31]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^30);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);