Finite Group Databasebeta

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#72259339

order23328
automorphism?
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

derived(C22⋊A4)⋊C27 ⋊ C2×C9

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenter(C22⋊A4)⋊C27C2×C9
FrattiniC3×C9C3×A4⋊S4
FittingC24×C9⋊C9C3⋊D3
socleC24×C3C3⋊(D9⋊C9)
prime radical#72229343C3
cosocle#72229223C2×C3
LCSG ▷ (C22⋊A4)⋊C27
UCSC1

897 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 897 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 35
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 35 points:

(1,4,3,2)(5,8,6,7)(10,26,35,23,29,16,27,22,31,13,17,28,15,34,25,24,14,30)(11,32,33,18,19,20)(12,21);(6,7,8)(9,17,24,19,35,30,18,10,22,12,29,23,33,25,16,20,15,34,21,14,13,11,31,28,32,27,26);(2,3,4)(6,7,8)

Polycyclic presentation:

2,3^6,2^4|b^3=c,e^3=f,f^3=g,d^a=d^2,e^a=e^2f^2g^2,e^b=efg^2,e^c=eg,f^a=f^2g^2,f^b=fg,g^a=g^2,h^d=i,h^e=i,i^a=hi,i^d=hi,i^e=hi,j^a=ijk,j^d=k,j^e=hij,k^a=hk,k^d=jk,k^e=hk

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

holomorph ofC22⋊C27

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