#72224516

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	C₃	((C₃⋊D₃)⋊C₃)⋊(C₃⋊D₃)
commutator, Fitting, cosocle, hypocenter	C₃×C₃²⋊H(3,𝔽₃)	C₂²
Frattini	C₃²	D₃×C₃⋊(C₃⋊D₃)
socle	C₃³	D₃×C₃⋊D₃
LCS	G ▷ C₃×C₃²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₃

1738 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1738 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 36

A minimal faithful permutation representation — 4 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224519, D₃×#11757730, #72226754, …
maximal normal subgroup of	#72224519, D₃×#11757730, #72226754, …
maximal quotient of	C₃×#72224516

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