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#72229397

order7776
automorphism?
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

derivedC22⋊(C22⋊C27) ⋊ C2×C9

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC22⋊(C22⋊C27)C2×C9
FrattiniC3×C9C3×#1020
FittingC24×C9⋊C9D3
socleC24×C3D9⋊C9
prime radical#72229320C3
cosocle#13048167C2×C3
LCSG ▷ C22⋊(C22⋊C27)
UCSC1

546 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 36 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 — 2 generators on 35 points:

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

Polycyclic presentation:

2,3^5,2^4|b^3=c,d^3=e,e^3=f,d^a=d^2e^2f^2,d^b=def^2,d^c=df,e^a=e^2f^2,e^b=ef,f^a=f^2,g^a=h,g^d=h,h^a=g,h^d=gh,i^a=hij,i^d=ghj,j^a=ghj,j^d=hij

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72259339

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