#72276228

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×C₉⋊C₉ ⋊ C₂
Sylow	C₉⋊C₉ ⋊ #127

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂³×C₉
Frattini, socle	C₂²×C₃²	C₂²×C₃×D₃
Fitting	C₂×D₄×C₉⋊C₉	C₂
prime radical	C₃×#63502	C₃
cosocle	C₂²×C₃×C₉	C₂³×C₃
hypercenter	C₂×C₃×D₄	AGL(1,ℤ/9)
hypocenter	C₉	C₉×#127
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×D₄

594 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 144

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

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

Polycyclic presentation:

2^3,3,2^2,3^3|c^2=f,d^3=g,h^3=i,b^a=be,c^a=cf,c^b=cf,h^b=h^2i^2,h^d=hi^2,i^b=i^2

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#127×D₉⋊C₉
maximal normal subgroup of	#127×D₉⋊C₉

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