Finite Group Databasebeta

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#72340827

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C2×#11714278
commutator, hypocenter#2679⋊C3C2×C4
FrattiniC22×C4C2×#1020
Fitting#7667013D3
socleC23C2×#5936
prime radical#72237719C2
cosocle#59816⋊C3C22
hypercenterC4#11714278
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2 ◁ C4

2008 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 2008 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 24
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

(1,8,3,10,7,2,6,5)(4,21,16,13)(9,20,14,24)(11,19,18,23)(12,15,22,17);(1,4,13,7,16,21)(2,14,12)(3,11,19,6,18,23)(5,15,20)(8,9,22)(10,17,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340829, #72340858
maximal normal subgroup of#72340829, #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, 2, 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, 2, 1, F.2*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
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.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*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.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.5*F.6*F.8*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.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.5*F.7);;
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, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.9*F.10);;
gap> SetConjugate(coll, 11, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);