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#72239827

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Splittings

Fitting#72239227 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#72209990
commutator, cosocle, hypocenter#72222569C22
Frattini, socleC25C2×#1020
Fitting#72239227D3
hypercenterC4#72209990
LCSG ▷ #72222569
UCSC1 ◁ C2 ◁ C4

7409 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 7409 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 96

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

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

Polycyclic presentation:

2^2,3,2^9|a^2=l,d^2=h,e^2=i,f^2=j,g^2=k,b^a=bhik,c^a=chijk,c^b=c^2,d^b=eh,d^c=eh,e^b=di,e^c=de,f^b=fg,f^c=gjk,f^d=fl,g^b=gk,g^c=fgj,g^d=gl,g^e=gl,g^f=gl,h^b=i,h^c=i,i^b=h,i^c=hi,j^b=jkl,j^c=k,k^c=jkl

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239833

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