#72228707

order	3888
automorphism	?
homology	Abelianization (H₁) = C₃⁴ · Schur multiplier (H₂) = C₂²×C₃⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃×C₉ ⋊ C₃²
Sylow	C₂⁴ ⋊ C₃×#737

Subgroups

	subgroup	G/H
center	C₉	C₃×A₄²
commutator, socle, cosocle	C₂⁴×C₃	C₃⁴
Frattini	C₃	C₃²×A₄²
Fitting	C₂⁴×C₃×C₉	C₃²
prime radical	(C₂²⋊A₄)⋊H(3,𝔽₃)	C₃
hypercenter	C₃×C₉	A₄²
hypocenter	C₂⁴	C₃×#737
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₉ ◁ C₃×C₉

1225 conjugacy classes of subgroups; only the 281 normal subgroups are shown.

normal subgroups only

This group has 1225 conjugacy classes of subgroups — too many to draw. Only its 281 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) = 35
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 35 points:

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

Polycyclic presentation:

3^5,2^4|d^3=e,c^b=ce,f^a=g,f^c=g,g^a=fg,g^c=fg,h^a=i,h^c=fgh,i^a=hi,i^c=fi

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

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