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#72235280

order5120
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

derived#17102⋊C5 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenter#17102⋊C5C4
FrattiniC25(C24⋊C5)⋊C2
Fitting#4122222D5
socleC24#66245
prime radical, cosocle#72235194C2
LCSG ▷ #17102⋊C5
UCSC1

1326 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 8 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) = 40
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 40 points:

(1,8,5,7)(2,4,9,14)(3,21,24,20)(6,31,18,29)(10,11,22,13)(12,38,23,37)(15,17,30,19)(16,40,34,36)(25,27)(26,32,39,33)(28,35);(1,11,23,26,30)(2,18,16,25,20)(3,12,36,15,5)(4,24,27,40,29)(6,32,35,10,9)(7,19,34,38,22)(8,17,33,37,21)(13,28,39,31,14)

Polycyclic presentation:

2^2,5,2^8|a^2=b,e^2=i,f^2=hij,c^a=c^4hi,c^b=chj,d^a=dhik,d^b=dk,d^c=fgjk,e^a=dehik,e^b=ehi,e^c=deghik,e^d=ej,f^a=dfh,f^b=fhij,f^c=dfk,f^d=fk,f^e=fh,g^a=defghik,g^b=ghj,g^c=egi,g^d=ghjk,g^e=gi,g^f=ghij,h^a=hjk,h^c=hij,i^a=ij,i^c=hk,j^c=h,k^c=i

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235285
maximal normal subgroup of#72235285

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