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#72227802

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC22#11714244
commutator, prime radical, cosocle, hypocenter#72227801C2
FrattiniC25#1020
Fitting#154071D3
socleC24#5934
hypercenterC23#85547
LCSG ▷ #72227801
UCSC1 ◁ C22 ◁ C23

2742 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 2742 conjugacy classes of subgroups — too many to draw. Only its 27 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) = 40
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2,3,2^9|a^2=j,c^2=g,d^2=h,b^a=b^2,c^a=cdhj,c^b=dgj,d^a=dh,d^b=cdj,d^c=dj,e^a=ef,e^b=f,e^c=ek,e^d=ei,f^b=ef,f^c=fik,f^d=fk,g^a=ghj,g^b=h,h^b=ghj,i^a=ik

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227822
maximal normal subgroup of#72227822

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