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#72284676

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72241342 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#346282C2×C3
FrattiniC25C2×C22⋊A4
Fitting#72241342C3
socleC22#46996⋊C3
LCSG ▷ #346282
UCSC1

1391 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 1391 conjugacy classes of subgroups — too many to draw. Only its 21 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:

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

Polycyclic presentation:

2,3,2^9|a^2=gk,c^2=g,d^2=gj,e^2=hj,f^2=hi,b^a=bik,c^a=cj,c^b=cd,d^a=dj,d^b=ck,d^c=dgjk,e^a=eghi,e^b=ef,e^c=egh,e^d=egi,f^a=fi,f^b=eghi,f^c=fgik,f^d=fghik,f^e=fgj,g^b=gk,g^e=gj,g^f=gjk,h^a=hk,h^b=gij,h^c=hj,h^d=hk,h^f=hj,i^a=ijk,i^b=ghik,i^c=ijk,i^d=ij,i^e=ijk,i^f=ij,j^b=k,k^b=jk

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72285257
maximal subgroup of#72284806, #72284920, #72285051
maximal normal subgroup of#72284806, #72284920, #72285051

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