Finite Group Databasebeta

← Prev (#72341985) Next (#72341987) →

#72341986

order3072
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

Fitting#6629044 ⋊ D3

Subgroups

subgroupG/H
centerC4#11713271
commutator, cosocle#5950C24
FrattiniC22×C4C23×S4
Fitting#6629044D3
socleC2C2×#11713271
hypercenterC2×C8#1020⋊C2
hypocenter#997C22×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

5964 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 5964 conjugacy classes of subgroups — too many to draw. Only its 125 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^5,3,2^5|d^2=ek,e^2=k,g^2=k,h^2=k,c^a=ck,d^a=de,e^a=ek,f^b=f^2,g^a=gijk,g^b=hk,g^f=h,h^a=hik,h^b=gk,h^f=ghk,h^g=hk,i^b=ij,i^f=j,i^g=ik,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

88 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, 4, F.5*F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 1, 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, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.8*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, 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);