Finite Group Databasebeta

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#72253509

order7680
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, derivedC5×#62817 ⋊ C2×C3
Hall#72253502 ⋊ C3
SylowC5 ⋊ #72209831

Subgroups

subgroupG/H
centerC2#72253500
commutator, Fitting, cosocle, hypocenterC5×#62817C2×C3
FrattiniC24D5×C22⋊A4
socleC2×C5#35346⋊C3
hypercenterC22C5⋊#85645
LCSG ▷ C5×#62817
UCSC1 ◁ C2 ◁ C22

3390 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 3390 conjugacy classes of subgroups — too many to draw. Only its 42 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)(6,7,10,13)(8,12,20,18,9,17,15,19,14,11,16,21);(1,2,4,5,3)(11,18)(12,19);(6,14)(7,17)(8,9)(10,20)(11,12)(13,21)(15,16)(18,19)

Polycyclic presentation:

2,3,2^6,5,2^2|a^2=k,c^b=cd,d^b=c,e^a=eh,e^b=ef,e^d=ej,f^a=fg,f^b=e,f^c=fj,g^b=h,g^c=gk,h^b=gh,h^d=hk,i^a=i^4,j^a=jk

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#72209831
maximal normal subgroup ofD5×#72209831

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