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#72226066

order2304
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socleC2C2×S42
commutator, cosocle, hypocenterΩ+(4,𝔽3)C23
Fitting#585D32
hypercenterC4S42
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C4

1437 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1437 conjugacy classes of subgroups — too many to draw. Only its 45 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:

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

2^3,3^2,2^5|a^2=j,c^2=j,f^2=j,g^2=j,c^a=cj,d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=fj,f^b=fj,f^d=fg,f^e=g,g^a=fgj,g^b=fgj,g^d=f,g^e=fg,g^f=gj,h^a=fh,h^b=fgi,h^d=fhj,h^e=i,h^f=hj,h^g=hj,i^a=ij,i^b=gh,i^d=gij,i^e=hi,i^g=ij

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226269, #72228062
maximal normal subgroup of#72226269, #72228062

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