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#72227671

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11715850
commutator, prime radical, cosocle, hypocenter#72227654C2
Frattini, hypercenterC23#87588
Fitting#8898087D3
LCSG ▷ #72227654
UCSC1 ◁ C22 ◁ C23

2170 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 2170 conjugacy classes of subgroups — too many to draw. Only its 26 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:

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

Polycyclic presentation:

2,3,2^9|a^2=jk,e^2=ij,f^2=ij,b^a=b^2,c^a=d,c^b=d,d^a=c,d^b=cd,e^a=cdeghjk,e^b=f,e^c=ei,e^d=eij,f^a=defgik,f^b=efij,f^c=fj,f^d=fi,f^e=fij,g^a=h,g^b=gh,g^c=gjk,g^e=gj,g^f=gk,h^a=g,h^b=g,h^d=hjk,h^e=hk,h^f=hjk,i^a=ik

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227696
maximal normal subgroup of#72227696

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