#72228645

order	5832
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₃)² ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	H(3,𝔽₃)²	C₂³
Frattini, socle	C₃²	#10620215
LCS	G ▷ H(3,𝔽₃)²
UCS	C₁

2070 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 2070 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 18
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 3 generators on 18 points:

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

Polycyclic presentation:

2^3,3^6|d^a=df,d^b=d^2,d^c=d^2,e^a=eg,e^b=e^2,e^c=de,f^a=f^2,f^b=f^2i,f^c=f^2i,f^d=fi,f^e=fi,g^a=g^2,g^b=g^2h,g^c=fgi,g^d=gi,g^e=gh,h^a=h^2,i^a=i^2

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228688, ((C₃⋊D₃)⋊C₃)⋊C₂², #72259881, …
maximal normal subgroup of	#72228688, ((C₃⋊D₃)⋊C₃)⋊C₂², #72259881, …

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