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#72268119

order6144
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268089 ⋊ C3
derived#72268079 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#72209717
commutator, cosocle, hypocenter#72268079C2×C3
FrattiniC23×Q8C2×C22⋊A4
Fitting#72268089C3
hypercenterC24#85645
LCSG ▷ #72268079
UCSC1 ◁ C22 ◁ C24

5884 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 5884 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 32
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2,3,2^10|c^2=k,d^2=k,g^2=l,h^2=l,c^a=cg,c^b=d,d^a=dh,d^b=cdk,d^c=dk,e^b=ef,e^d=ei,f^b=e,f^c=fi,f^d=fi,g^a=gl,g^b=h,g^c=gl,g^d=gij,g^f=gkl,h^a=hl,h^b=ghij,h^c=hijl,h^d=hl,h^e=hkl,h^f=hkl,h^g=hl,i^a=ikl,j^a=jk

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268126
maximal normal subgroup of#72268126

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