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#72259104

order4752
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Splittings

Fitting, socleC32×C11 ⋊ GL(2,𝔽3)
derivedC11×PSU(3,𝔽2)⋊C3 ⋊ C2
SylowC11 ⋊ AGL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC11×PSU(3,𝔽2)⋊C3C2
Fitting, socleC32×C11GL(2,𝔽3)
LCSG ▷ C11×PSU(3,𝔽2)⋊C3
UCSC1

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 99

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

2,3,11,2^3,3^2|d^2=f,e^2=f,b^a=b^2,c^a=c^10,d^a=de,d^b=e,e^a=ef,e^b=def,e^d=ef,g^a=g^2,g^d=h,g^e=gh,g^f=g^2,h^a=g^2h,h^b=gh,h^d=g^2,h^e=gh^2,h^f=h^2

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72259128, D11×AGL(2,𝔽3)
maximal normal subgroup of#72259128, D11×AGL(2,𝔽3)

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