#72230587

order	10584
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃³×C₇² ⋊ C₂³
Sylow	C₇² ⋊ C₂×D₃×C₃⋊D₃
Sylow	C₃³ ⋊ C₂×D₇²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³×C₇²	C₂³
LCS	G ▷ C₃³×C₇²
UCS	C₁

1348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 126

Polycyclic presentation:

2^3,3^3,7^2|d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^b=f^2,g^a=g^6,g^c=g^6,h^a=g^2h,h^c=h^6

Conjugacy classes & character table

279 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230589, D₃×#72213616
maximal normal subgroup of	#72230589, D₃×#72213616

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