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#72239165

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C8

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72239117 ⋊ C3
derived#184421 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2#72209989
commutator, cosocle, hypocenter#184421C2×C3
FrattiniC2×C42C2×C22⋊A4
Fitting#72239117C3
socleC23C2×C22⋊(C42⋊C3)
LCSG ▷ #184421
UCSC1 ◁ C2

2105 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 2105 conjugacy classes of subgroups — too many to draw. Only its 34 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2,3,2^9|c^2=g,d^2=h,e^2=k,f^2=jk,g^2=j,h^2=k,b^a=bj,c^a=cj,c^b=dgijk,d^a=dk,d^b=cdi,d^c=di,e^b=fk,e^c=ejk,e^d=eik,f^b=ef,f^c=fik,f^d=fj,g^b=hj,h^b=ghi,j^b=k,k^b=jk

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239179

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