Finite Group Databasebeta

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#72341879

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#10585848 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#72341874
commutator#85666C2×C4
FrattiniC2×C8C2×#1020
Fitting#10585848D3
prime radical#72341689C2
cosocleC2×#85666C22
hypercenterM32#1020
hypocenter#997#102
LCSG ▷ #85666 ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

2978 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 2978 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

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,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, 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, 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);