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#72230720

order4096
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72224639
commutatorC32C22×C32
Frattini, cosocle#93666C23
prime radicalC2×D64C16
LCSG ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #7052 ◁ #72225140 ◁ G
LECSG ▷ #93666 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

295 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 96
transitive degreeμt(G) = 2048

A minimal faithful permutation representation — 3 generators on 96 points:

(1,2,4,9,5,10,17,25,6,11,18,16,20,27,32,31,7,12,19,26,21,28,13,23,22,29,30,3,8,14,24,15)(34,40,42,56,43,57,62,60,44,59,67,87,68,61,92,88)(35,45,47,70,48,71,74,73)(36,49,51,78)(37,52)(38,54)(41,55,63,58,64,82,93,90,65,83,66,91,89,85,86,84)(46,69,75,72,76,95,96,94)(50,77,80,79)(53,81);(1,3,13,12,20,25,4,14,22,26,32,11,5,15,30,28,7,16,17,2,8,23,19,27,6,9,24,29,21,31,18,10)(34,41,62,93,68,89,42,63,44,65,92,86,43,64,67,66)(35,46,74,96,48,76,47,75)(36,50,51,80)(37,53)(40,55,60,90,61,85,56,58,59,83,88,84,57,82,87,91)(45,69,73,94,71,95,70,72)(49,77,78,79)(52,81);(33,34,35,40,36,41,45,58,37,42,46,60,49,66,72,55,38,43,47,61,50,64,73,84,52,62,75,56,79,86,69,85,39,44,48,59,51,65,71,91,53,67,76,88,78,63,94,83,54,68,74,57,80,89,70,90,81,92,96,87,77,93,95,82)

Polycyclic presentation:

2^12|a^2=d,b^2=dfj,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,c^a=ce,c^b=cg,d^c=dikl,e^a=eg,e^b=ei,e^d=ek,f^c=fkl,f^e=fl,g^a=gi,g^b=gk,g^d=gl,h^c=hl,i^a=ik,i^b=il,k^a=kl

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72230814
maximal normal subgroup of#72230814
maximal quotient of#72230814
automorphism group ofSD256

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.6*F.10);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);