Finite Group Databasebeta

← Prev (#72252410) Next (#72252412) →

#72252411

order12288
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#72241399 ⋊ D3
derived#72227582 ⋊ C2×C4

Subgroups

subgroupG/H
center, socleC2#72218442
commutator, hypocenter#72227582C2×C4
FrattiniC2×H(7,𝔽2)C2×S4
Fitting#72241399D3
prime radical#72241383C2
cosocle#72227585C22
hypercenterC23#72215879
LCSG ▷ #72227582
UCSC1 ◁ C2 ◁ C22 ◁ C23

6886 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 6886 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,3,15,32,6,13,17,18)(2,28,20,16)(4,5,14,26,19,21,22,31)(7,9)(8,27)(10,29)(11,30,12,24)(23,25);(1,2)(3,9)(4,16)(5,20,21,11)(6,12)(7,22,23,14)(8,26,27,31)(10,17)(13,25)(15,29)(18,28,32,24)(19,30)

Polycyclic presentation:

2^3,3,2^9|a^2=c,e^2=l,f^2=l,g^2=m,h^2=m,d^a=d^2,d^b=d^2,e^a=egklm,e^b=el,e^c=eil,e^d=f,f^a=efghk,f^b=ef,f^c=fijl,f^d=efl,f^e=fl,g^a=gi,g^b=gm,g^d=h,g^f=gk,h^a=ghjm,h^b=gh,h^d=ghm,h^e=hk,h^g=hm,i^d=ij,i^f=il,i^h=im,j^a=ij,j^b=ij,j^d=i,j^e=jl,j^f=jl,j^g=jm,j^h=jm,k^a=kl,k^c=km,l^a=lm

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267084
maximal normal subgroup of#72267084

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 5, F.12);;
gap> SetPower(coll, 6, F.12);;
gap> SetPower(coll, 7, F.13);;
gap> SetPower(coll, 8, F.13);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.11*F.12*F.13);;
gap> SetConjugate(coll, 5, 2, F.5*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.12);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.5*F.6*F.7*F.8*F.11);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10*F.12);;
gap> SetConjugate(coll, 6, 4, F.5*F.6*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.13);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.10*F.13);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.13);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.13);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.12);;
gap> SetConjugate(coll, 9, 8, F.9*F.13);;
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);;
gap> SetConjugate(coll, 10, 5, F.10*F.12);;
gap> SetConjugate(coll, 10, 6, F.10*F.12);;
gap> SetConjugate(coll, 10, 7, F.10*F.13);;
gap> SetConjugate(coll, 10, 8, F.10*F.13);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);