Finite Group Databasebeta

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#72246787

order15360
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

FittingC5×#62817 ⋊ Dic3
derivedC5×#11710730 ⋊ C4
SylowC5 ⋊ #72245113

Subgroups

subgroupG/H
center, Frattini, hypercenterC22#72246556
commutator, hypocenterC5×#11710730C4
FittingC5×#62817Dic3
socleC22×C5#11715843
prime radical, cosocleD5×#11710730C2
LCSG ▷ C5×#11710730
UCSC1 ◁ C22

23250 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 23250 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(2,3,4,5)(7,10,11,13)(8,12)(15,21)(18,19);(1,2,5,3,4)(7,8,10)(11,12,13)(15,16,18)(19,20,21);(6,10)(7,8)(9,13)(11,12)(14,18)(15,16)(17,21)(19,20)

Polycyclic presentation:

2^2,3,5,2^8|a^2=b,c^a=c^2,d^a=d^3,d^b=d^4,e^a=ef,e^c=f,f^c=ef,g^a=gh,g^c=h,g^f=gl,h^c=gh,h^e=hl,i^a=ij,i^c=j,i^f=ik,j^c=ij,j^e=jk

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#72209641
maximal normal subgroup ofAGL(1,𝔽5)×#72209641

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^3);;
gap> SetConjugate(coll, 4, 2, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);