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#72228157

order9216
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

Fitting#153709 ⋊ C3⋊D3
derived#72222575 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC2#72214357
commutator, cosocle, hypocenter#72222575C2
Frattini, socleC25A4⋊S4
Fitting#153709C3⋊D3
LCSG ▷ #72222575
UCSC1 ◁ C2

1424 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 72

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2,3^2,2^9|d^2=h,e^2=i,f^2=j,g^2=k,b^a=b^2,c^a=c^2,d^a=dh,d^b=de,d^c=ehi,e^a=deh,e^b=dil,e^c=dehl,e^d=el,f^a=dfij,f^b=gj,f^c=efhi,f^d=fl,f^e=fl,g^a=efgh,g^b=fg,g^c=dgil,g^d=gl,h^b=hil,h^c=i,i^a=hil,i^b=h,i^c=hil,j^a=hjl,j^b=k,j^c=ijl,k^a=ijkl,k^b=jk,k^c=hkl

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

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", "k", "l");;
gap> coll := SingleCollector(F, [2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.4*F.9*F.12);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.8*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 3, F.4*F.7*F.9*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.12);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.12);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.8*F.10*F.12);;
gap> SetConjugate(coll, 10, 2, F.11);;
gap> SetConjugate(coll, 10, 3, F.9*F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 11, 2, F.10*F.11);;
gap> SetConjugate(coll, 11, 3, F.8*F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);