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#72323170

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72286339
commutatorC2×C32C22×C8
Frattini, cosocle#7330C23
prime radical#33981C8
LCSG ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×#1311 ◁ #149495 ◁ G
LECSG ▷ #7330 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

414 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 128
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 128 points:

(1,2)(3,17,19,66,21,68,5,31)(4,24,26,87,28,51,98,65,30,89,100,106,45,90,41,118)(6,36,38,112)(7,40)(8,44)(9,16)(10,22,12,34,49,81,58,76)(11,27,52,94,53,73,122,97,55,95,116,108,64,25,62,83)(13,39,61,115)(14,42)(15,46)(18,70,71,111,72,126,93,54,74,123,103,86,33,60,75,124)(20,78,32,107)(23,85,77,69,82,127,35,110)(29,47,79,88,92,57,37,113,105,119,99,128,101,121,114,120)(43,67)(48,80,56,104)(50,91,59,84,117,109,125,102)(63,96);(1,3,6,22,7,23,39,84,8,19,20,76,42,77,80,91,9,21,38,81,43,82,115,102,46,5,32,34,96,35,104,109)(2,10,13,17,14,50,36,69,15,12,56,31,40,59,107,85,16,49,61,68,63,117,112,110,44,58,48,66,67,125,78,127)(4,25,29,93,26,94,99,72,28,95,92,71,98,83,114,18,30,73,105,75,100,108,79,33,45,27,101,103,41,97,37,74)(11,51,54,113,52,106,70,57,53,24,86,88,122,65,126,47,55,90,124,120,116,87,123,121,64,89,111,128,62,118,60,119);(1,4)(2,11)(3,18)(5,33)(6,37)(7,41)(8,45)(9,30)(10,47)(12,57)(13,60)(14,62)(15,64)(16,55)(17,65)(19,72)(20,79)(21,74)(22,83)(23,71)(24,69)(25,91)(26,96)(27,102)(28,46)(29,104)(31,106)(32,99)(34,108)(35,75)(36,111)(38,114)(39,101)(40,116)(42,100)(43,98)(44,53)(48,70)(49,119)(50,88)(51,85)(52,67)(54,78)(56,123)(58,121)(59,113)(61,126)(63,122)(66,87)(68,118)(73,109)(76,94)(77,93)(80,105)(81,97)(82,103)(84,95)(86,112)(89,110)(90,127)(92,115)(107,124)(117,128)(120,125)

Polycyclic presentation:

2^11|a^2=d,b^2=g,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=bgh,c^a=ce,c^b=cgk,d^b=djk,d^c=dijk,e^a=eg,e^b=ek,e^c=egijk,e^d=ej,f^b=fk,f^c=fjk,f^e=fk,g^a=gi,g^c=gijk,g^d=gk,h^c=hk,i^a=ij,i^c=ijk,j^a=jk,j^c=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72323172
maximal normal subgroup of#72323172

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