Finite Group Databasebeta

← Prev (#72261831) Next (#72261833) →

#72261832

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#10995
commutatorC22×C4⋊C4C22×C4
Frattini, cosocleC22×C4×D4C23
prime radical#946373C2
LCSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4×D4 ▷ C23 ▷ C2 ▷ C1

4096 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 4096 conjugacy classes of subgroups — too many to draw. Only its 157 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) = 24
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

(2,5)(6,7)(11,19)(12,13)(15,22)(16,17)(20,21)(23,24);(2,6)(3,8)(5,7)(9,10)(11,15,21,23)(12,17)(13,16)(14,18)(19,24,20,22);(1,2,3,5)(4,7,8,6)(9,11,14,21)(10,15)(12,20,13,19)(16,24)(17,22)(18,23)

Polycyclic presentation:

2^10|b^2=g,c^2=dfi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=cd,c^b=cdei,d^a=di,d^b=dh,d^c=di,e^a=ehj,e^b=ehij,e^c=ehij,e^d=ej,f^a=fi,f^c=fi,g^c=gi,h^c=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72261838, #72262104, #72262111
maximal normal subgroup of#72261838, #72262104, #72262111

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