Finite Group Databasebeta

← Prev (#72277387) Next (#72277389) →

#72277388

order5760
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

SylowC5 ⋊ #983⋊A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D5×A4×S4
commutator, cosocle, hypocenterC5×#994C22×C3
FittingC5×H(5,𝔽2)C2×C3×D3
socleC2×C5C2×A4×S4
LCSG ▷ C5×#994
UCSC1 ◁ C2

768 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 53
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 2 generators on 53 points:

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

Polycyclic presentation:

2^2,3,2^2,3,5,2^3|d^2=j,e^2=j,h^2=j,i^2=j,b^a=bj,d^c=e,e^c=dej,e^d=ej,f^a=f^2,g^b=g^4,h^a=hj,h^f=i,i^a=hi,i^f=hij,i^h=ij

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

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