Finite Group Databasebeta

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#72338971

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237414
commutatorC22×H(5,𝔽2)C22×C4
Frattini, cosocle#35527C23
prime radical#6628901C22
LCSG ▷ C22×H(5,𝔽2) ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35527 ◁ G
LECSG ▷ #35527 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1

5668 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 5668 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(2,7,10,22,11,24,18,23)(4,16,17,20)(9,25,26,32,12,21,27,31)(13,28)(14,19)(29,30);(1,2,8,10,6,11,5,18)(3,12,19,27,15,9,14,26)(4,7,16,22,17,24,20,23)(13,25,28,31,29,21,30,32);(1,3)(2,9)(4,13,17,29)(5,19)(6,15)(7,21,24,25)(8,14)(10,27)(11,12)(16,30,20,28)(18,26)(22,31,23,32)

Polycyclic presentation:

2^11|a^2=e,b^2=ghk,c^2=hj,e^2=j,g^2=k,b^a=bd,c^a=cf,c^b=cj,d^a=dg,d^b=dij,d^c=dik,e^b=egk,e^c=eh,e^d=eik,f^a=fh,f^b=fik,f^c=fj,f^e=fj,g^a=gi,g^b=gk,g^c=gk,g^d=gk,g^f=gk,h^a=hj,h^b=hk,h^d=hk,i^b=ik,i^c=ik,j^b=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338974, #72338976, #72338987, …
maximal normal subgroup of#72338974, #72338976, #72338987, …

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