#72258708

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#616936 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#126⋊C₃
commutator, cosocle	C₂²×C₄²	C₂³×C₃
Frattini, socle	C₂⁴	C₂³×A₄
Fitting	#616936	C₃
hypercenter	C₄⋊C₄	#126⋊C₃
hypocenter	C₄²	C₃×#121
LCS	G ▷ C₂²×C₄² ▷ C₄²
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

1189 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 1189 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 96

Polycyclic presentation:

2^3,3,2^6|a^2=e,c^2=e,g^2=i,h^2=j,b^a=be,c^a=cf,c^b=cef,d^a=di,d^c=di,g^b=gij,g^d=hi,h^b=hi,h^d=gh,i^d=j,j^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258724
maximal normal subgroup of	#72258724

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);