Finite Group Databasebeta

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#72271679

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#412 ⋊ D3
derivedQ8⋊SL(2,𝔽3) ⋊ C23

Subgroups

subgroupG/H
center, Frattini, socleC22C22×#1020
commutator, cosocle, hypocenterQ8⋊SL(2,𝔽3)C23
Fitting#412D3
hypercenterC42#1020
LCSG ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42

2300 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 2300 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 192

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^3,3,2^6|b^2=j,c^2=i,e^2=i,f^2=i,g^2=ij,h^2=ij,b^a=bj,c^a=ci,d^a=d^2,e^a=fi,e^d=f,f^a=ei,f^d=efi,f^e=fi,g^a=gh,g^d=h,h^a=hij,h^d=ghij,h^g=hij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271691, #72271698
maximal normal subgroup of#72271691, #72271698

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