#72243411

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

C₃²×#6891 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃×C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))
commutator, cosocle	C₃⁵	C₂×C₃²
Frattini	C₃²	C₃²×C₃⋊(C₃⋊D₃)
Fitting	C₃²×#6891	C₂
socle	C₃⁴	C₃²×D₃
prime radical	C₃²×C₃⋊(C₃⋊(C₃⋊D₃))	C₃
hypercenter	C₃²	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))
hypocenter	C₃⁴	C₂×3₋¹⁺²
LCS	G ▷ C₃⁵ ▷ C₃⁴
UCS	C₁ ◁ C₃ ◁ C₃²

13240 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 13240 conjugacy classes of subgroups — too many to draw. Only its 269 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) = 24
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243422, #72253800, C₃⋊D₃×#92259
maximal normal subgroup of	#72243422, #72253800, C₃⋊D₃×#92259

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