Finite Group Databasebeta

← Prev (#72341887) Next (#72341889) →

#72341888

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72341884
commutator#85666C2×C4
FrattiniC23×C8C2×S4
Fitting#6629044D3
prime radical#72341679C2
cosocleC2×#85666C22
hypercenterC2×C8#1020⋊C2
hypocenter#997#102
LCSG ▷ #85666 ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

1544 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1544 conjugacy classes of subgroups — too many to draw. Only its 45 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 — 2 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

70 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, 1, F.3);;
gap> SetPower(coll, 2, F.3*F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 2, F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
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);