#72277625

order	2952
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃²×C₄₁ ⋊ C₈
Sylow	C₄₁ ⋊ C₃⋊(C₃⋊C₈)
Sylow	C₃² ⋊ #66502

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₄₁	C₈
prime radical	C₃²×D₄₁	C₄
cosocle	C₃²×#4223	C₂
LCS	G ▷ C₃²×C₄₁
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 369

Polycyclic presentation:

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

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#66502, #72277636, D₃×C₃⋊#66502
maximal normal subgroup of	C₃⋊D₃×#66502, #72277636, D₃×C₃⋊#66502

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