#72342749

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting

#10586984 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₈	C₂²×#1020
commutator, hypocenter	#997	C₂³×C₄
Frattini	C₄	C₂³×#1020
Fitting	#10586984	D₃
socle	C₂	C₂²×C₄×#1020
prime radical	#72233470	C₂
cosocle	#5950	C₂⁴
hypercenter	#131	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₈ ◁ #131

23826 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 23826 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 64

Polycyclic presentation:

2^5,3,2^5|a^2=e,e^2=k,d^c=dk,f^b=f^2,g^f=h,h^b=gh,h^f=gh,i^b=j,i^f=j,i^g=ik,j^b=i,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342856, #72342925
maximal normal subgroup of	#72342856

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