#72277709

order	14760
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃²×C₄₁ ⋊ C₂×C₄×C₅
Hall	#72277660 ⋊ C₅
Hall	C₃²×C₄₁⋊C₅ ⋊ C₂×C₄
Hall	C₄₁⋊C₅ ⋊ #624
Sylow	C₄₁ ⋊ C₅×#624
Sylow	C₃² ⋊ C₂×#4223⋊C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₄₁	C₂×C₄×C₅
prime radical	C₃⋊D₃×D₄₁⋊C₅	C₂
cosocle	C₃²×D₄₁	C₂²×C₅
LCS	G ▷ C₃²×C₄₁
UCS	C₁

148 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 246

Polycyclic presentation:

2^2,5,2,3^2,41|a^2=d,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2,g^a=g^9,g^c=g^18,g^d=g^40

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃²×#4223⋊C₅
maximal normal subgroup of	D₃²×#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", "g");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 3, 3, 41]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^9);;
gap> SetConjugate(coll, 7, 3, F.7^18);;
gap> SetConjugate(coll, 7, 4, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);