#72224526

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

C₃×C₃²⋊H(3,𝔽₃) ⋊ C₂²

Subgroups

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

2297 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 2297 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11757749, #72226753, #72226754
maximal normal subgroup of	D₃×#11757749, #72226753, #72226754

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