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#72267833

order10240
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting#72267796 ⋊ D5
derived#72267832 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC22#72226628
commutator, cosocle, hypocenter#72267832C2
Frattini, socleC26(C24⋊C5)⋊C2
Fitting#72267796D5
LCSG ▷ #72267832
UCSC1 ◁ C22

4090 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 4090 conjugacy classes of subgroups — too many to draw. Only its 13 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) = 72
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 72 points:

(1,7,4,2,6,8,16,11)(3,25,14,24,15,10,18,23)(5,28,12,17,21,9,19,22)(13,26,30,29,20,32,31,27)(33,45,37,39)(34,36)(35,55,44,53)(38,48,51,61)(40,47)(41,46,54,43)(42,67,59,56)(49,57)(50,65,52,62)(58,72,60,68)(63,71,66,70)(64,69);(2,8,15,20,12)(3,13,19,11,7)(4,17,5,18,23)(9,26,30,27,25)(10,28,32,31,29)(14,24,16,22,21)(33,35,42,57,48)(34,38,49,56,41)(36,46,58,71,61)(37,43,59,69,62)(39,51,66,72,54)(40,52,64,67,55)(44,60,70,65,47)(45,50,63,68,53)

Polycyclic presentation:

2,5,2^10|c^2=k,d^2=ikl,e^2=g,f^2=g,b^a=b^4,c^a=cdfgijl,c^b=cf,d^a=efgik,d^b=cdi,d^c=di,e^a=ehijkl,e^b=deikl,e^c=ek,e^d=eikl,f^a=dei,f^b=eghkl,f^c=fgh,f^d=fgij,f^e=fghj,h^a=ghil,h^b=hjk,i^a=gl,i^b=hj,j^a=hk,j^b=ghijl,k^a=ghijl,k^b=hk,l^a=gi,l^b=jkl

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267843

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