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#72227809

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72227807 ⋊ C3
derived#355894 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2×C4#85647
commutator, cosocle, hypocenter#355894C2×C3
FrattiniC23×C4C2×C22⋊A4
Fitting#72227807C3
socleC22#11709856
LCSG ▷ #355894
UCSC1 ◁ C2×C4

6476 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 6476 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 24
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

(2,5,3)(6,7,8)(10,16,11,19,12,23)(17,24,21,22,18,20);(2,6)(3,8)(9,22,14,23,15,17,13,11)(10,20,18,12,19,21,24,16);(1,7,4,5)(2,3,6,8)(9,21,14,12,15,20,13,16)(10,23,18,22,19,11,24,17)

Polycyclic presentation:

2,3,2^9|c^2=g,d^2=h,e^2=hj,f^2=ghij,g^2=k,h^2=k,i^2=k,b^a=bgh,c^a=cgi,c^b=dgik,d^a=dhi,d^b=cdi,d^c=di,e^a=ehi,e^b=fhi,e^c=eij,e^d=ej,f^a=fgh,f^b=efijk,f^c=fik,f^d=fijk,f^e=fijk,g^b=hk,g^d=gk,g^e=gk,g^f=gk,h^b=ghik,h^c=hk,h^f=hk

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227818
maximal normal subgroup of#72227818

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