Finite Group Databasebeta

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#72258556

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#72233407 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#72252244
commutator, cosocle#72215518C22
FrattiniC23×C4C2×#1020
Fitting#72233407D3
hypercenterC2×C4#87588⋊C2
hypocenter#11710730D4
LCSG ▷ #72215518 ▷ #11710730
UCSC1 ◁ C2 ◁ C2×C4

At least 6063 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has at least 6063 conjugacy classes of subgroups — too many to draw. Only its 36 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:

2^3,3,2^8|c^2=l,g^2=k,h^2=k,b^a=bc,c^a=cl,c^b=cl,d^a=d^2,d^b=d^2,e^a=ef,e^b=efj,e^d=f,f^b=fi,f^d=ef,g^a=fg,g^b=fgijl,g^d=h,g^e=gk,h^a=eghk,h^b=eghjkl,h^d=ghk,h^e=hk,h^f=hk,h^g=hk,i^d=j,i^h=il,j^a=ij,j^b=ij,j^d=ij,j^g=jl,k^b=kl

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258595
maximal normal subgroup of#72258595

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.12);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.12);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.7*F.9*F.10*F.12);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.5*F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.5*F.7*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);