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#72325872

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237487
commutatorC22×H(5,𝔽2)C22×C4
Frattini, cosocle#35346C23
prime radical#72237724C2
LCSG ▷ C22×H(5,𝔽2) ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #35346 ◁ G
LECSG ▷ #35346 ▷ C25 ▷ C23 ▷ C2 ▷ C1

4527 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 4527 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^11|a^2=i,b^2=e,c^2=hj,b^a=bd,c^a=chk,c^b=cf,d^b=dg,d^c=dijk,e^a=eg,e^c=eh,e^d=ei,f^a=fi,f^b=fh,f^c=fj,f^e=fj,g^b=gi,g^c=gk,g^f=gk,h^a=hk,h^b=hj,h^d=hk,i^c=ik,j^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72326041, #72337732, #72337734
maximal normal subgroup of#72326041, #72337732, #72337734

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