#72277532

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#10624943 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₉	C₃⋊D₃²
commutator, cosocle, hypocenter	H(5,𝔽₃)	C₂²×C₃
Frattini, socle	C₃	C₃×C₃⋊D₃²
Fitting	#10624943	C₂²
prime radical	#11758087	C₃
LCS	G ▷ H(5,𝔽₃)
UCS	C₁ ◁ C₉

1597 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 1597 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 81

Polycyclic presentation:

2^2,3^6|c^3=h,d^b=d^2,e^a=e^2,e^b=e^2,f^b=f^2,f^d=fh,g^a=g^2,g^b=g^2,g^e=gh^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72277535

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 3, F.8);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);