#72276118

order	2592
automorphism	?
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Sylow

H(5,𝔽₂) ⋊ C₉⋊C₉

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃×C₂²⋊A₄
commutator	C₃×H(5,𝔽₂)	C₃×C₉
Fitting	C₃×C₉×H(5,𝔽₂)	C₃
prime radical, cosocle	C₃²×H(5,𝔽₂)	C₃²
hypercenter	C₂×C₃×C₉	C₂²⋊A₄
hypocenter	H(5,𝔽₂)	C₉⋊C₉
LCS	G ▷ C₃×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×C₉

427 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 26
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 3 generators on 26 points:

(2,5,3)(6,7,8)(18,19,22,20,21,25,23,24,26);(2,6)(3,8)(9,10,12,11,13,15,14,16,17)(19,21,24)(22,26,25);(1,5,4,7)(2,8,6,3)

Polycyclic presentation:

3^4,2^5|a^3=c,b^3=d,e^2=i,f^2=i,b^a=bd,e^b=f,f^b=efi,f^e=fi,g^b=h,g^f=gi,h^b=gh,h^e=hi

Conjugacy classes & character table

201 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₉⋊C₉×#997, #72276184
maximal normal subgroup of	C₉⋊C₉×#997, #72276184
maximal quotient of	#72276144

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