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#72239354

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72239321 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72209996
commutator, Fitting, cosocle, hypocenter#72239321C3
FrattiniC22×C43C22⋊A4
socleC23#72209995
LCSG ▷ #72239321
UCSC1 ◁ C2

6768 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 6768 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 112
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 112 points:

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

Polycyclic presentation:

3,2^12|d^2=f,e^2=g,h^2=k,i^2=l,j^2=m,b^a=c,c^a=bc,d^a=eflm,d^b=di,d^c=dh,e^a=delm,e^b=ehijl,e^c=ehilm,e^d=elm,f^a=g,f^b=fm,f^c=fk,g^a=fglm,g^b=gk,g^c=gkm,h^a=jl,h^b=hl,h^c=hk,h^e=hlm,i^a=hiklm,i^b=il,i^c=il,i^d=ilm,j^a=hjl,j^b=jklm,j^c=jkl,j^d=jlm,k^a=m,l^a=kl,m^a=km

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239373

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