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#72228062

order4608
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

FittingC2×#585 ⋊ D32

Subgroups

subgroupG/H
centerC4C2×S42
commutator, cosocle, hypocenterΩ+(4,𝔽3)C24
Frattini, socleC2C22×S42
FittingC2×#585D32
hypercenterC2×C4S42
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

9757 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 9757 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 96 points:

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

Polycyclic presentation:

2^4,3^2,2^5|a^2=k,d^2=k,g^2=k,h^2=k,c^a=ck,d^a=dk,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.11);;
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, 4, 1, F.4*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);