Finite Group Databasebeta

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#72230814

order8192
nilpotency class7
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72230720
commutatorC64C22×C32
Frattini, cosocle#72230688C23
prime radical#153706C16
LCSG ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ #93666 ◁ #72230692 ◁ G
LECSG ▷ #72230688 ▷ #7329 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

318 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^13|a^2=e,b^2=d,c^2=dfj,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,l^2=m,b^a=begiklm,c^a=cgiklm,d^a=diklm,e^b=eg,e^c=ei,e^d=ek,f^a=fkl,f^e=flm,g^b=gi,g^c=gk,g^d=gl,g^f=gm,h^a=hlm,h^e=hm,i^b=ik,i^c=il,i^d=im,j^a=jm,k^b=kl,k^c=km,l^b=lm

Conjugacy classes & character table

191 conjugacy classes — view the full character table.

Relationships

automorphism group ofD128, Q256
holomorph ofC128

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.6*F.10);;
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> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7*F.9*F.11*F.12*F.13);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9*F.11*F.12*F.13);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.11*F.12*F.13);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.12*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.13);;
gap> SetConjugate(coll, 8, 1, F.8*F.12*F.13);;
gap> SetConjugate(coll, 8, 5, F.8*F.13);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 9, 4, F.9*F.13);;
gap> SetConjugate(coll, 10, 1, F.10*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);