Finite Group Databasebeta

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#72239927

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C2×#85547
commutator, prime radical, cosocle, hypocenterC22⋊(#95⋊C3)C22
FrattiniC23C2×#1020
FittingC22×#376D3
socleC25C2×S4
hypercenterC4#85547
LCSG ▷ C22⋊(#95⋊C3)
UCSC1 ◁ C2 ◁ C4

1136 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 48 points:

(1,13,20,14,6,34,5,3)(2,22,28,11,10,30,9,7)(4,40,19,36)(8,46,27,43)(12,39,32,16)(15,17)(18,37)(21,45,42,24)(23,25)(26,35)(29,44,47,41)(31,33,38,48);(1,8,31,10,4,24,6,27,16,2,19,41)(3,29,37,7,12,35,14,21,15,11,33,23)(5,26,39,28,18,44,20,25,38,9,17,45)(13,47,36,22,32,43,34,42,40,30,48,46)

Polycyclic presentation:

2^2,3,2^7|a^2=j,b^2=j,d^2=h,e^2=i,b^a=bj,c^b=c^2,d^b=ehij,d^c=ehj,e^b=dhij,e^c=dej,e^d=ej,f^b=fg,f^c=g,g^c=fg,h^b=i,h^c=i,i^b=h,i^c=hij

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239930

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