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#72341933

order3072
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72320061 ⋊ C3

Subgroups

subgroupG/H
center, socleC2C2×#32930⋊C3
commutator, cosocle#3573C23×C3
FrattiniC22×C8C23×A4
Fitting#72320061C3
hypercenterM32#120⋊C3
hypocenterH(5,𝔽2)C2×C3×D8
LCSG ▷ #3573 ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

2423 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 2423 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)(5,6,23)(8,24)(11,25)(12,33,40,32,13,38)(15,16,47)(18,19,50)(20,22,21)(26,34)(27,39,56,37,28,55)(29,30,58)(35,52,62,51,36,61)(41,42,49)(43,46,57)(44,45,48)(53,60,64,59,54,63);(1,38,31,64,4,61,3,55,7,40,9,63,14,62,17,56)(2,57,34,50,11,47,8,21,10,58,26,49,25,44,24,23)(5,33,43,54,18,52,15,39,20,13,29,60,41,36,45,28)(6,12,46,59,19,35,16,27,22,32,30,53,42,51,48,37);(2,8)(3,9)(4,14)(5,20)(6,22)(10,24)(11,34)(12,37)(13,39)(15,43)(16,46)(17,31)(21,23)(25,26)(27,32)(28,33)(29,45)(30,48)(35,53)(36,54)(38,56)(40,55)(44,58)(47,57)(51,59)(52,60)(61,63)(62,64)

Polycyclic presentation:

2^3,3,2^7|b^2=ef,e^2=fk,f^2=k,g^2=k,h^2=k,b^a=be,c^b=ck,e^a=ef,f^a=fk,g^a=gj,g^d=gh,h^a=hi,h^d=g,h^g=hk,i^d=j,i^g=ik,j^d=ijk,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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.11);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);