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#72271983

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#87524
commutator, cosocle#3529⋊C3C23
FrattiniC2×C4C22×#1020
Fitting#10586807D3
prime radical#72268457C2
hypercenter#124#1020
hypocenter#5465C2×D4
LCSG ▷ #3529⋊C3 ▷ #5465
UCSC1 ◁ C22 ◁ C42 ◁ #124

7599 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 7599 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 64
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,30,20,11,17,14,6,31,5,3,19,26)(2,32,18,12,15,22,10,29,9,7,24,28)(4,13,16,25)(8,21,23,27)(34,40)(36,49,47)(37,50,48)(39,41)(42,43,58,53,54,59)(45,61,63)(46,62,60)(51,52,57,55,56,64);(1,2,6,10)(3,12,11,7)(4,15)(5,9,20,18)(8,19)(13,29)(14,28,26,22)(16,24)(17,23)(21,30)(25,32)(27,31)(33,47,49,36,38,50,37,48)(34,57,54,51,40,64,43,55)(35,62,61,60,44,63,46,45)(39,58,56,42,41,59,52,53);(3,11)(7,12)(13,25)(14,26)(21,27)(22,28)(29,32)(30,31)(33,34)(35,39)(36,42)(37,43)(38,40)(41,44)(45,51)(46,52)(47,58)(48,53)(49,54)(50,59)(55,60)(56,61)(57,63)(62,64)

Polycyclic presentation:

2^4,3,2^6|b^2=jk,c^2=jk,d^2=j,f^2=j,g^2=j,h^2=jk,i^2=jk,b^a=bd,c^a=ck,c^b=cjk,d^a=dj,d^b=dj,e^b=e^2,f^b=fg,f^e=g,g^b=gj,g^e=fgj,g^f=gj,h^b=hi,h^e=i,h^g=hk,i^b=ijk,i^e=hijk,i^f=ik,i^h=ijk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272066
maximal normal subgroup of#72272066

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