#72259647

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

Aliases

No aliases for this group.

Splittings

Sylow

C₂⁴ ⋊ 3₋¹⁺⁴

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃²×A₄²
commutator, socle, cosocle	C₂⁴×C₃	C₃⁴
Fitting	C₂⁴×3₋¹⁺²	C₃²
prime radical	C₃×(C₃×A₄)⋊A₄	C₃
hypercenter	3₋¹⁺²	A₄²
hypocenter	C₂⁴	3₋¹⁺⁴
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

1188 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 1188 conjugacy classes of subgroups — too many to draw. Only its 278 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 108

Polycyclic presentation:

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

Conjugacy classes & character table

160 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, 4, 1, F.4*F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 3, F.6*F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);