Finite Group Databasebeta

← Prev (#72245688) Next (#72245690) →

#72245689

order10368
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC4#72213207
commutator, cosocle, hypocenter#997⋊C9C22×C3
Frattini, socleC2×C3C2×C3×A4⋊S4
FittingC32×#585C3⋊D3
LCSG ▷ #997⋊C9
UCSC1 ◁ C4

1490 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 1490 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 57
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 57 points:

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

Polycyclic presentation:

2^2,3^4,2^5|a^2=k,e^3=f,g^2=k,h^2=k,d^b=d^2,e^b=e^2f^2,e^c=ef,f^b=f^2,g^b=gk,g^d=h,g^e=h,h^b=gh,h^d=ghk,h^e=ghk,h^g=hk,i^b=hijk,i^d=j,i^e=ghi,i^h=ik,j^b=gjk,j^d=ij,j^e=gjk,j^g=jk

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268997
maximal normal subgroup of#72268997

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