#72276455

order	3996
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃²×C₃₇ ⋊ C₃×C₄
Hall	C₃²×C₃₇⋊C₃ ⋊ C₄
Sylow	C₃₇ ⋊ C₃×C₃⋊Dic₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₃₇	C₃×C₄
prime radical	C₃²×D₃₇⋊C₃	C₂
cosocle	C₃²×D₃₇	C₂×C₃
LCS	G ▷ C₃²×C₃₇
UCS	C₁

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 333

Polycyclic presentation:

2,3,2,3^2,37|a^2=c,d^a=d^2,e^a=e^2,f^a=f^31,f^b=f^10,f^c=f^36

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#3866⋊C₃, #72276564
maximal normal subgroup of	C₃⋊D₃×#3866⋊C₃, #72276564

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, 3, 2, 3, 3, 37]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^31);;
gap> SetConjugate(coll, 6, 2, F.6^10);;
gap> SetConjugate(coll, 6, 3, F.6^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);