#72268831

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72268824 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂³	C₂²×#120⋊C₃
commutator, cosocle	C₂⁴×Q₈	C₂³×C₃
Frattini, socle	C₂⁵	C₂³×A₄
Fitting	#72268824	C₃
hypercenter	C₂²×D₄	#120⋊C₃
hypocenter	C₂²×Q₈	C₃×#120
LCS	G ▷ C₂⁴×Q₈ ▷ C₂²×Q₈
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

74450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 74450 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 192

Polycyclic presentation:

2^3,3,2^7|a^2=k,c^2=f,g^2=k,h^2=k,b^a=bk,c^a=ce,c^b=cf,g^c=gj,g^d=gh,h^c=hi,h^d=g,h^g=hk,i^d=j,j^d=ij

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268887, #72268929
maximal normal subgroup of	#72268929

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.11);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);