#72224535

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃×C₃²⋊H(3,𝔽₃) ⋊ C₂²
derived	C₃²×H(3,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×H(3,𝔽₃)	C₂²×C₃
Frattini	C₃²	C₃×#1113
Fitting	C₃×C₃²⋊H(3,𝔽₃)	C₂²
socle	C₃³	C₃×D₃²
LCS	G ▷ C₃²×H(3,𝔽₃)
UCS	C₁

1168 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 21 points:

(2,3)(4,5,6,8,10,9)(7,12)(13,14,16,18,15,19)(17,21);(1,2,3)(5,7)(6,10)(8,11)(9,12)(14,17)(18,21)(19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224536, D₃×(((C₃⋊D₃)⋊C₃)⋊D₃)⋊C₃, #72226753
maximal normal subgroup of	#72224536, D₃×(((C₃⋊D₃)⋊C₃)⋊D₃)⋊C₃, #72226753

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