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#72235395

order2048
nilpotency class4
automorphism?
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C43

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#110180
commutatorC22×C42C2×C42
Frattini, cosocleC22×H(3,𝔽4)C23
prime radical#5035253C22
LCSG ▷ C22×C42 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×H(3,𝔽4) ◁ G
LECSG ▷ C22×H(3,𝔽4) ▷ C24 ▷ C22 ▷ C1

2751 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 2751 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,10,6,11,28,41)(3,14,17,8,13,29,42,15)(4,19,22,55,26,56,36,59)(7,31,33,37,30,12,21,44)(9,35,20,58,39,54,52,63)(16,48,51,23,38,62,40,32)(18,46,53,47,45,60,50,34)(24,43,27,61,25,57,49,64);(1,3,15,2)(4,20,25,40)(5,10,28,41)(6,13,8,11)(7,18,34,31)(9,36)(12,30,45,47)(14,17,29,42)(16,49)(19,54,56,35)(21,37)(22,39)(23,61,58,55)(24,51,26,52)(27,38)(32,64,63,59)(33,44)(43,62,57,48)(46,50)(53,60);(1,4,21,54)(2,9,37,55)(3,16,50,61)(5,27,34,58)(6,26,33,35)(7,32,29,36)(8,25,60,48)(10,40,45,56)(11,39,44,59)(12,43,42,20)(13,38,53,64)(14,22,30,23)(15,24,46,62)(17,52,31,57)(18,19,41,51)(28,49,47,63)

Polycyclic presentation:

2^11|a^2=d,b^2=ej,c^2=dfgjk,d^2=k,f^2=k,b^a=bde,c^a=cf,c^b=cdejk,d^b=di,d^c=dh,e^a=eijk,e^c=eijk,e^d=ej,f^a=fhk,f^b=fh,f^c=fijk,f^d=fk,f^e=fk,g^a=ghijk,g^c=ghijk,g^d=gjk,g^f=gj,h^a=hk,h^b=hk,h^c=hj,i^a=ij,i^b=ij,i^c=ijk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235400
maximal normal subgroup of#72235400

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