Finite Group Databasebeta

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#72253021

order5184
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socleC2×C3C3×A4×S4
commutator, cosocleC3×#994C2×C32
FittingC32×H(5,𝔽2)C3×D3
prime radicalC32×#5925C3
hypercenterC2×C32A4×S4
hypocenter#994C2×3−1+2
LCSG ▷ C3×#994 ▷ #994
UCSC1 ◁ C2×C3 ◁ C2×C32

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

normal subgroups only

This group has 519 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) = 57
transitive degreeμt(G) = 288

A minimal faithful permutation representation — 2 generators on 57 points:

(1,40,19,36,18,42,7,32,6,29,21,48,9,31,25,11,3,46,5,47,27,35,16,43)(2,38,45,8,34,37,12,14,24,20,39,26,13,15,41,22,10,30,28,17,44,4,33,23)(49,52,57,51,55,53,54,50,56);(1,37,18,8,6,21,9,30,3,22,27,16)(2,33,44,29,10,43,13,39,24,35,34,48)(4,23,17,7,19,15,20,26,14,5,25,38)(11,32,45,28,31,42,36,47,41,12,40,46)(50,52,55)(53,57,56)

Polycyclic presentation:

2,3^3,2^2,3,2^3|a^2=j,c^3=d,e^2=j,f^2=j,h^2=j,i^2=j,c^b=cd,e^c=ef,f^c=e,f^e=fj,g^a=g^2,h^a=hj,h^g=i,i^a=hi,i^g=hij,i^h=ij

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72253023

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