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#72244010

order26244
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72221882 ⋊ C22
derived#14554857 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#14554857C22×C3
FrattiniC33C3×C3⋊#1113
Fitting#72221882C22
socleC3C3×#72214385
LCSG ▷ #14554857
UCSC1

6773 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 6773 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 81
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 3 generators on 81 points:

(1,80,20,60,21,81,6,61,16,28,17,77,5,63,42,58,4,75)(2,48,32,50,27,73,8,36,31,13,18,69,11,70,43,37,10,62)(3,46,39,30,40,72,15,29,49,9,19,64,22,59,44,26,14,65)(7,66,57,41,52,47,23,34,55,12,45,67,33,68,71,35,25,38)(24,74,76,53,56,79,51,54,78);(1,2,7)(3,13,24,12,9,28)(4,43,45,16,18,71)(5,8,33,6,11,23)(10,55,21,31,25,20)(14,62,56,38,29,77)(15,50,51,41,26,60)(17,32,52,42,27,57)(19,73,54,47,46,75)(22,37,53,35,30,58)(34,65,63,49,36,76)(39,70,79,68,64,80)(40,69,74,67,59,81)(44,48,78,66,72,61);(2,8,11)(7,23,33)(10,27,18)(24,53,51)(25,52,45)(28,60,58)(31,43,32)(54,74,56)(55,71,57)(61,80,63)(75,77,81)(76,79,78)

Polycyclic presentation:

2^2,3^8|g^3=j^2,d^a=d^2,e^b=e^2,e^d=eh,f^a=e^2f^2,f^b=f^2,f^d=fi,g^a=g^2j,g^c=gj^2,g^e=gh^2,g^f=gh^2,h^a=h^2,h^b=h^2j,h^e=hj,h^f=hj,i^a=hij,i^b=i^2,i^e=ij,j^a=j^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72253869
maximal normal subgroup of#72253869

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