#72243418

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14552182 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#13091503
commutator, cosocle	C₃⁵	C₂×C₃²
Frattini, socle	C₃³	C₃²×C₃⋊D₃
Fitting	#14552182	C₂
prime radical	C₃×#92301	C₃
hypocenter	C₃⁴	C₂×3₋¹⁺²
LCS	G ▷ C₃⁵ ▷ C₃⁴
UCS	C₁ ◁ C₃

1330 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 1330 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 27
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 27 points:

(2,7,9,4,5,8)(3,6)(11,18,15,12,16,17)(13,14)(19,25,23,22,20,24,26,21,27);(1,2,4)(3,5,8)(6,9,7)(10,12,16)(11,13,18)(14,17,15)(19,21,27,22,25,23,26,20,24);(10,11,12)(13,15,18)(14,16,17)

Polycyclic presentation:

2,3^7|b^3=d,c^3=d,c^b=cd,e^a=e^2,e^b=eg,e^c=eg^2,f^a=f^2,f^b=fg^2,f^c=fg^2h,g^a=g^2,h^a=h^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243422, #72269289
maximal normal subgroup of	#72243422, #72269289

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