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#72226075

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

1423 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1423 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:

(1,81,73,22,41,55,9,53,26,3,24,94)(2,11,10,16)(4,18,70,57,5,93,28,20,72,17,7,91)(6,87,25,21,66,61,8,90,34,19,69,54)(12,80,44,48,47,79,35,76,46,37,43,77)(13,51,14,49,33,50,29,40,36,38,15,39)(23,86,71,52,64,92,42,85,82,62,65,96)(27,59,67,60,31,95,32,58,74,56,30,88)(45,78,75,83)(63,89,68,84);(2,10)(4,24)(5,7)(6,31)(8,30)(11,16)(12,42)(14,33)(15,36)(17,53)(18,20)(19,59)(21,58)(23,35)(25,69)(26,72)(27,74)(28,41)(32,67)(34,66)(37,52)(39,51)(40,50)(43,63)(44,82)(45,64)(46,71)(47,68)(48,62)(54,90)(55,93)(56,95)(57,81)(60,88)(61,87)(65,75)(70,73)(76,89)(77,92)(78,86)(79,96)(80,84)(83,85)(91,94);(1,2,9,10)(3,16,22,11)(4,23,28,42)(5,29,7,13)(6,15,8,33)(12,41,35,24)(14,30,36,31)(17,52,57,62)(18,38,20,49)(19,51,21,40)(25,68,34,63)(26,71,73,82)(27,65,32,64)(37,53,48,81)(39,59,50,58)(43,66,47,69)(44,70,46,72)(45,67,75,74)(54,89,61,84)(55,92,94,96)(56,86,60,85)(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,b^a=bj,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#72226297, #72228062
maximal normal subgroup of#72226297, #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, 2, 1, F.2*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);