Finite Group Databasebeta

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#72340812

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#52301 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#85558
commutator, cosocle, hypocenter#2679⋊C3C22
Frattini, socleC23C2×#1020
Fitting#52301D3
hypercenterC4#85558
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2 ◁ C4

701 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 701 conjugacy classes of subgroups — too many to draw. Only its 19 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) = 24
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

(1,14,9,13)(2,8,16,19)(3,6,15,7)(4,12,5,17)(10,24,11,23,21,22,18,20);(1,9,2,4,7,6,5,16)(3,24,15,11)(8,23,19,10)(12,21,17,20)(13,18,14,22)

Polycyclic presentation:

2^2,3,2^7|a^2=hij,d^2=i,e^2=ij,b^a=bij,c^a=c^2hi,c^b=c^2,d^a=dehij,d^b=dehij,d^c=ehi,e^a=eij,e^b=eij,e^c=deh,e^d=eh,f^a=deghj,f^b=deghj,f^c=fg,f^d=fi,f^e=fhj,g^a=dfhij,g^b=dfhij,g^c=f,g^d=gij,g^e=ghi,i^a=hj,i^b=hj,i^c=ij,j^a=hi,j^b=hi,j^c=hi

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340814, #72340836, #72340845
maximal normal subgroup of#72340814, #72340836, #72340845

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