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#72342007

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#10585848 ⋊ D3

Subgroups

subgroupG/H
center, socleC2C2×#11715730
commutator, cosocle#85666C23
FrattiniC8C22×#1020
Fitting#10585848D3
hypercenterM32#1020
hypocenter#997C2×D8
LCSG ▷ #85666 ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

4150 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 4150 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 64

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

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

Polycyclic presentation:

2^5,3,2^5|b^2=de,d^2=ek,e^2=k,b^a=bd,c^b=ck,d^a=de,e^a=ek,f^a=f^2,g^a=h,g^f=h,h^a=g,h^f=gh,i^a=ij,i^f=j,i^g=ik,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

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