Finite Group Databasebeta

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#72247659

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#10586097 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#72216669
commutator, cosocleC2×#5950C23
FrattiniC2×C4C22×#1020
Fitting#10586097D3
hypercenter#127#1020
hypocenter#997#120
LCSG ▷ C2×#5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

16602 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 16602 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(3,8)(4,20,16)(5,19,17)(7,12)(10,18,15)(11,27,25)(13,14,31,23,24,26)(21,22,32,28,29,30);(1,15,14,21)(2,26,29,17)(3,30,20,10)(4,9,31,22)(5,25,8,32)(6,27,24,28)(7,19,11,13)(12,16,18,23);(2,7,9,12)(10,21,25,28)(11,22,18,29)(15,32,27,30)

Polycyclic presentation:

2^5,3,2^5|a^2=e,b^2=dek,d^2=k,g^2=k,h^2=k,b^a=bd,c^a=ce,c^b=cek,d^b=dk,d^c=dk,e^b=ek,f^b=f^2,g^b=gh,g^f=h,h^b=hk,h^f=ghk,h^g=hk,i^b=j,i^f=j,i^g=ik,i^h=ik,j^b=i,j^f=ij,j^h=jk

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252511, #72252541, #72267089
maximal normal subgroup of#72252511, #72252541, #72267089

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