#72222824

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72222731 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂³	C₂×#560⋊C₃
commutator, cosocle	C₂²×#95	C₂³×C₃
Frattini, socle	C₂⁵	C₂³×A₄
Fitting	#72222731	C₃
hypercenter	C₂²×C₄	#560⋊C₃
hypocenter	#95	C₃×#120
LCS	G ▷ C₂²×#95 ▷ #95
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

7127 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 7127 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^7|c^2=fk,g^2=i,h^2=j,c^a=ce,c^b=cf,d^c=dik,g^a=gjk,g^b=gik,g^d=hik,h^a=hij,h^b=hjk,h^d=ghk,h^g=hk,i^d=j,j^d=ijk

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227335
maximal normal subgroup of	#72227335

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