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#72226197

order4608
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, FrattiniC4C2×S42
commutator, hypocenterΩ+(4,𝔽3)C22×C4
FittingC2×#585D32
socleC2C2×#11923458
prime radicalC2×#11923520C2
cosocle(#41⋊C3)⋊A4C23
hypercenterC2×C4S42
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

2831 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 2831 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 96
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 96 points:

(1,64,25)(2,10)(3,75,50)(4,5,69)(6,62,33)(7,67,27)(8,66,24)(9,23,70)(11,34)(12,42,46,35,47,43)(13,15,36,28,32,14)(16,17,85)(18,81,57)(19,88,52)(20,83,49)(21,41,87)(22,61,84,58,60,68)(26,30,71)(29,63,31)(37,76,79,73,80,77)(38,40,74,53,56,39)(44,72)(45,86,51)(48,92,96,89,91,94)(54,82,55)(59,65)(78,95)(90,93);(4,23)(5,7)(6,30)(8,29)(12,22)(13,28)(14,15)(16,41)(17,19)(18,45)(20,54)(24,66)(25,69)(26,71)(27,64)(31,63)(32,36)(33,62)(35,58)(37,48)(38,53)(39,40)(42,65)(43,68)(44,61)(46,84)(47,59)(49,83)(50,85)(51,86)(52,75)(55,82)(56,74)(57,81)(60,72)(67,70)(73,89)(76,93)(77,94)(78,92)(79,96)(80,90)(87,88)(91,95);(1,2,3,11,9,10,21,34)(4,22,16,48,27,58,52,89)(5,28,17,53,7,13,19,38)(6,15,18,40,8,32,20,56)(12,41,37,64,35,75,73,23)(14,45,39,29,36,54,74,30)(24,65,49,93,33,59,57,90)(25,68,50,94,70,84,87,96)(26,61,51,92,31,60,55,91)(42,83,76,62,47,81,80,66)(43,85,77,67,46,88,79,69)(44,86,78,63,72,82,95,71)

Polycyclic presentation:

2^4,3^2,2^5|a^2=d,d^2=k,g^2=k,h^2=k,c^a=ck,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2,g^a=gk,g^b=gk,g^e=gh,g^f=h,h^a=ghk,h^b=ghk,h^e=g,h^f=gh,h^g=hk,i^a=gi,i^b=ghj,i^e=gik,i^f=j,i^g=ik,i^h=ik,j^a=jk,j^b=hi,j^e=hjk,j^f=ij,j^h=jk

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

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