#72245451

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	3₋¹⁺²×H(5,𝔽₂) ⋊ D₃
derived	C₉×#997 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72220104
commutator, cosocle, hypocenter	C₉×#997	C₂×C₃
Frattini, socle	C₂×C₃	C₃×C₃⋊#1020
Fitting	3₋¹⁺²×H(5,𝔽₂)	D₃
LCS	G ▷ C₉×#997
UCS	C₁ ◁ C₂

895 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

2,3^4,2^5|c^3=e,c^a=c^2e^2,c^b=ce^2,d^a=d^2,e^a=e^2,f^a=g,f^d=g,g^a=f,g^d=fg,h^a=i,h^d=i,h^g=hj,i^a=h,i^d=hi,i^f=ij

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#5934
maximal normal subgroup of	AGL(1,ℤ/9)×#5934

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", "j");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.5^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);