Finite Group Databasebeta

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#72340815

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC4#11714278
commutator, hypocenter#2679⋊C3C2×C4
FrattiniC22×C4C2×#1020
Fitting#7667013D3
socleC23C2×#5936
prime radical#72237719C2
cosocle#59816⋊C3C22
LCSG ▷ #2679⋊C3
UCSC1 ◁ C4

2014 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 2014 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 48 points:

(1,29,38,3,19,32)(2,42,35,24,28,46)(4,13,26,12,20,48)(5,41,16,34,39,30)(6,33,18,14,15,47)(7,25,44,10,40,36)(8,43,37,23,27,31)(9,45,11,22,17,21);(1,11,4,2)(3,23,30,24)(5,35)(6,8,16,10)(7,14,21,12)(9,18)(13,46)(15,17)(19,37)(20,36)(22,32,43,47)(25,26,28,39)(27,38)(29,45)(31,33)(34,44)(40,41)(42,48)

Polycyclic presentation:

2^3,3,2^7|a^2=c,c^2=i,e^2=j,f^2=jk,d^a=d^2,e^a=ef,e^b=eij,e^c=eij,e^d=fij,f^a=fik,f^b=fijk,f^c=fijk,f^d=efi,f^e=fi,g^a=efhjk,g^b=gijk,g^c=gijk,g^d=gh,g^e=gj,g^f=gik,h^a=eg,h^b=hij,h^c=hij,h^d=g,h^e=hjk,h^f=hij,j^a=ik,j^d=jk,k^a=ij,k^d=ij

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340816, #72340821, #72340858
maximal normal subgroup of#72340816, #72340821, #72340858

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