#72244608

order	3280
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₄₁ ⋊ C₄×C₅
Hall	C₂²⋊#4223 ⋊ C₅
Hall	C₄₁⋊C₅ ⋊ C₂²⋊C₄
Sylow	C₄₁ ⋊ C₅×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#4223⋊C₅
commutator, socle	C₂×C₄₁	C₂×C₄×C₅
Fitting	C₂²×C₄₁	C₄×C₅
prime radical	C₂²×D₄₁⋊C₅	C₂
cosocle	C₂×D₄₁	C₂²×C₅
hypercenter	C₂²	#4223⋊C₅
hypocenter	C₄₁	C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂ ◁ C₂²

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 164

Polycyclic presentation:

2^2,5,2^2,41|a^2=d,b^a=bde,f^a=f^9,f^c=f^18,f^d=f^40,f^e=f^40

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#4223⋊C₅, #72254112
maximal normal subgroup of	D₄×#4223⋊C₅

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, 41]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^9);;
gap> SetConjugate(coll, 6, 3, F.6^18);;
gap> SetConjugate(coll, 6, 4, F.6^40);;
gap> SetConjugate(coll, 6, 5, F.6^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);