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#72245149

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

Aliases

No aliases for this group.

Splittings

Fitting#19753 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#1455⋊C3C22
FrattiniC2×C42C2×S4
Fitting#19753D3
socleC22#73084
hypocenterC82⋊C3D4
LCSG ▷ #1455⋊C3 ▷ C82⋊C3
UCSC1

295 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 15 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:

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

Polycyclic presentation:

2^3,3,2^6|a^2=c,e^2=g,f^2=h,g^2=i,h^2=j,b^a=bc,d^a=d^2,e^a=egij,e^b=egj,e^c=ei,e^d=fgij,f^a=efi,f^b=fhij,f^c=fj,f^d=ef,g^a=gi,g^b=gi,g^d=hi,h^a=gh,h^b=hj,h^d=gh,i^d=j,j^a=ij,j^d=ij

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245194
maximal normal subgroup of#72245194

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