#72277687

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃²⋊H(5,𝔽₃) ⋊ C₂²
derived	C₃×C₃²⋊H(3,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#72221945
commutator, cosocle, hypocenter	C₃×C₃²⋊H(3,𝔽₃)	C₂²×C₃
Frattini, socle	C₃²	C₃×C₃⋊#1113
Fitting	C₃²⋊H(5,𝔽₃)	C₂²
LCS	G ▷ C₃×C₃²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₃

4509 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 4509 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 36
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 36 points:

(2,5,6)(4,8,9)(10,11,13,15,21,17)(12,35,19,34,20,27)(14,24)(16,30)(18,25)(23,26,31,36,32,33)(28,29);(2,4)(5,9)(6,8)(10,12,18)(11,23,27,14,16,26)(13,20,29)(15,32,34,24,25,36)(17,31,35,22,28,33)(19,30,21);(1,2,4)(3,6,8)(5,9,7)

Polycyclic presentation:

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

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277883
maximal normal subgroup of	#72277883

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