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#72237384

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Splittings

Fitting#5024691 ⋊ D3
derived#11709219 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#11709219C22
Frattini, socleC24C2×#1020
Fitting#5024691D3
LCSG ▷ #11709219
UCSC1

3082 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 3082 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^2,3,2^8|b^2=jk,d^2=h,e^2=i,b^a=bjk,c^a=cjk,c^b=c^2j,d^a=dhk,d^b=eh,d^c=eh,e^a=eij,e^b=di,e^c=de,f^a=fjk,f^c=g,f^d=fj,f^e=fk,g^a=gk,g^b=fg,g^c=fg,g^d=gk,g^e=gjk,h^b=i,h^c=i,i^b=h,i^c=hi,j^b=k,j^c=jk,k^b=j,k^c=j

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237390
maximal normal subgroup of#72237390

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