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#72268972

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

Aliases

No aliases for this group.

Splittings

Fitting#10586807 ⋊ D3

Subgroups

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

8037 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 8037 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) = 48
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^4,3,2^6|b^2=dj,c^2=k,d^2=j,f^2=j,g^2=j,b^a=bd,c^a=cjk,c^b=cj,d^a=dj,e^b=e^2,f^b=fg,f^e=g,g^b=gj,g^e=fgj,g^f=gj,h^b=fghijk,h^e=i,h^f=hj,h^g=hjk,i^b=gik,i^e=hi,i^f=ik,i^g=ij

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72269004
maximal normal subgroup of#72269004

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