#72253809

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃×H(3,𝔽₃)×3₋¹⁺² ⋊ C₂²
derived	C₃×C₉×H(3,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₉×H(3,𝔽₃)	C₂²×C₃
Frattini	C₃²	C₃×C₃⋊#1113
Fitting	C₃×H(3,𝔽₃)×3₋¹⁺²	C₂²
socle	C₃³	C₃×D₃×C₃⋊D₃
LCS	G ▷ C₃×C₉×H(3,𝔽₃)
UCS	C₁

4077 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 4077 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 162

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72253827

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