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#72341744

order3072
automorphism?
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72319553 ⋊ C3

Subgroups

subgroupG/H
centerC4C42×C22⋊A4
commutator#585C3×C42
FrattiniC2×C8C22×C22⋊A4
Fitting#72319553C3
socleC2#97×C22⋊A4
prime radicalC2×#5950C2×C4
cosocleC2×#3573C22×C3
hypercenter#347C22⋊A4
hypocenterH(5,𝔽2)C3×#97
LCSG ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C4 ◁ #347

5861 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 5861 conjugacy classes of subgroups — too many to draw. Only its 145 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:

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

Polycyclic presentation:

2^2,3,2^8|a^2=d,b^2=e,e^2=f,f^2=k,g^2=k,h^2=k,b^a=bf,d^b=dk,e^a=ek,g^c=gh,h^c=g,h^g=hk,i^c=ij,i^g=ik,i^h=ik,j^c=i,j^h=jk

Conjugacy classes & character table

166 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, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
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.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);