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#72267870

order12288
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267867 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72226653
commutator, cosocle, hypocenter#72267865C2×C3
Frattini#94849C2×A4
Fitting#72267867C3
socleC23#72226651
prime radical#72267869C2
LCSG ▷ #72267865
UCSC1 ◁ C2

3253 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 3253 conjugacy classes of subgroups — too many to draw. Only its 17 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) = 96

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2,3,2^11|a^2=m,c^2=g,d^2=gi,e^2=hijkl,f^2=ijl,g^2=m,b^a=bhim,c^a=cefijl,c^b=dhkl,d^a=de,d^b=cdgh,d^c=dgh,e^a=ehijkl,e^b=fgijkm,e^c=eghij,e^d=ehiklm,f^a=fim,f^b=efghkm,f^c=fgijkl,f^d=fgjklm,f^e=fjlm,g^a=gjm,g^b=gil,g^d=gk,g^e=gkl,g^f=glm,h^b=hik,h^c=hkm,h^d=hl,i^b=hklm,i^c=iklm,i^d=ik,j^c=jkm,j^d=jkl,k^b=klm,l^b=k

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267881

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