Finite Group Databasebeta

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#72224594

order56448
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC24×C72 ⋊ C22×C3×D3
derivedC72×C22⋊A4 ⋊ C23×C3
SylowC72 ⋊ C2×C3×#1020⋊C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC72×C22⋊A4C23×C3
FrattiniC22S4×#64423⋊C2
FittingC24×C72C22×C3×D3
socleC22×C72C22×C3×S4
LCSG ▷ C72×C22⋊A4
UCSC1

5250 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 5250 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 22
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 3 generators on 22 points:

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

Polycyclic presentation:

2^3,3^2,7^2,2^4|e^b=e^2,f^a=f^6,f^c=f^6,f^d=f^4,g^a=f^2g,g^c=g^6,g^d=g^4,h^a=hjk,h^b=hi,h^e=i,i^a=ij,i^e=hi,j^e=k,k^b=jk,k^e=jk

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

holomorph ofC22×D7

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