Finite Group Databasebeta

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#72340819

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

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 595 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) = 48
transitive degreeμt(G) = 48

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

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

Polycyclic presentation:

2^2,3,2^7|a^2=h,b^2=h,d^2=i,e^2=ij,b^a=bij,c^a=c^2,c^b=chij,d^a=dehi,d^c=ehi,e^a=eh,e^c=deh,e^d=eh,f^a=degij,f^c=fg,f^d=fi,f^e=fhj,g^a=df,g^c=f,g^d=gij,g^e=ghi,i^a=hj,i^c=ij,j^a=hi,j^c=hi

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340820, #72340836, #72340845
maximal normal subgroup of#72340820, #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);;
gap> SetPower(coll, 2, F.8);;
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);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
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.9*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);;
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, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);