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#72245146

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

Aliases

No aliases for this group.

Splittings

Fitting#17339 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#1455⋊C3C22
FrattiniC2×C42C2×S4
Fitting#17339D3
socleC22#73048
hypocenterC82⋊C3D4
LCSG ▷ #1455⋊C3 ▷ C82⋊C3
UCSC1

264 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 16 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:

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

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=eij,e^c=ei,e^d=fgij,f^a=efi,f^b=fi,f^c=fj,f^d=ef,g^a=gi,g^d=hi,h^a=gh,h^d=gh,i^d=j,j^a=ij,j^d=ij

Conjugacy classes & character table

33 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.9*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.9);;
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, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
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);