Finite Group Databasebeta

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#72258808

order4608
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24×C3

Aliases

No aliases for this group.

Splittings

FittingC24×D8 ⋊ C3⋊D3

Subgroups

subgroupG/H
centerC2D4×A4⋊S4
commutator, cosocleC4×A42C23
FrattiniC4C22×A4⋊S4
FittingC24×D8C3⋊D3
socleC25D4×C3⋊D3
hypercenterD8A4⋊S4
hypocenterA42AGL(1,ℤ/8)
LCSG ▷ C4×A42 ▷ C2×A42 ▷ A42
UCSC1 ◁ C2 ◁ C4 ◁ D8

4150 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 4150 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 16
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 16 points:

(1,4,3,2)(5,8,6,7)(10,14)(13,15);(6,7,8)(10,13)(11,16)(14,15);(2,3,4)(6,7,8)(9,10,11,15,12,14,16,13)

Polycyclic presentation:

2^5,3^2,2^4|a^2=d,d^2=e,b^a=bd,c^a=ce,d^b=de,f^c=f^2,g^c=g^2,h^f=i,h^g=i,i^c=hi,i^f=hi,i^g=hi,j^c=ijk,j^f=k,j^g=hij,k^c=hk,k^f=jk,k^g=hk

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

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