Finite Group Databasebeta

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#72283861

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237777
commutator#1781C22×C4
Frattini, cosocle#20586C23
prime radical#72283821C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1853 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1853 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^11|b^2=ghk,c^2=efhijk,d^2=h,f^2=hijk,b^a=bghk,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dh,e^a=egk,e^b=ei,e^c=egijk,e^d=eij,f^a=fgk,f^b=fg,f^c=fgijk,f^d=fij,f^e=fij,g^b=gjk,g^c=gi,g^d=gjk,g^f=gj,h^b=hjk,h^e=hk,h^f=hk,i^a=ij,i^b=ik,i^c=ij,i^d=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285103, #72285850
maximal normal subgroup of#72285103, #72285850

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