#72254074

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃²×#6900 ⋊ C₂²
derived	C₃²×C₉² ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₉²	C₂²×C₃
Frattini	C₃²	C₃×C₃⋊#1113
Fitting	C₃²×#6900	C₂²
socle	C₃⁴	C₃×D₃²
LCS	G ▷ C₃²×C₉²
UCS	C₁

3038 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 3038 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) = 24
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 24 points:

(4,5,6)(8,14,15,10,11,12)(9,13)(16,21,20,23,19,17)(22,24);(2,3)(5,6)(7,11,10,9,15,12,13,8,14)(17,18)(19,20)(21,24)(22,23);(1,2,3)

Polycyclic presentation:

2^2,3^7|e^3=h,f^3=i,d^a=d^2,e^a=e^2h^2,e^b=e^2h^2,e^c=eh,f^a=e^2f,f^b=f^2i^2,f^c=fi,g^a=g^2,h^a=h^2,h^b=h^2,i^a=h^2i,i^b=i^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254082, C₃⋊D₃×#92265⋊C₂, D₃×#72214299
maximal normal subgroup of	#72254082, C₃⋊D₃×#92265⋊C₂, D₃×#72214299

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.9^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);