Finite Group Databasebeta

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#72310331

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2#72224760
commutator#3549⋊C3C2×C4
FrattiniC24C2×#1020
Fitting#7627400D3
socleC23#87524
prime radical, cosocleC2×#3549⋊C3C22
hypocenter#5464C4⋊C4
LCSG ▷ #3549⋊C3 ▷ #5464
UCSC1 ◁ C2

2652 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 2652 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 28
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 28 points:

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

Polycyclic presentation:

2^4,3,2^6|a^2=c,b^2=d,f^2=j,g^2=k,h^2=k,i^2=jk,b^a=bd,e^a=ej,e^b=e^2k,e^d=ek,f^a=fj,f^b=gj,f^e=gj,g^a=gk,g^b=fk,g^e=fg,h^a=hjk,h^b=hi,h^d=hk,h^e=ik,h^f=hjk,h^g=hk,i^a=ij,i^b=ij,i^d=ijk,i^e=hi,i^f=ik,i^g=ij,j^b=k,j^e=k,k^b=j,k^e=jk

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#72224760
maximal normal subgroup ofC4×#72224760

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