Finite Group Databasebeta

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#72340825

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

Aliases

No aliases for this group.

Splittings

Fitting#60045 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#11715758
commutator, cosocle, hypocenter#2679⋊C3C22
Frattini, socleC23C2×#1020
Fitting#60045D3
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2

635 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 40
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340826, #72340845, #72340851
maximal normal subgroup of#72340826, #72340845, #72340851

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