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#72228370

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

Aliases

No aliases for this group.

Splittings

FittingC2×H(5,𝔽2) ⋊ D32

Subgroups

subgroupG/H
center, Frattini, socleC2C2×S42
commutator, cosocle, hypocenterΩ+(4,𝔽3)C23
FittingC2×H(5,𝔽2)D32
hypercenterC22S42
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C22

1432 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1432 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,52,21,6)(2,61,38,31)(3,49,5,20)(4,22,46,18)(7,48,17,24)(8,63,34,12)(9,62,11,42)(10,57,60,55)(13,50,23,27)(14,53,45,43)(15,25,19,47)(16,51,44,26)(28,64,30,37)(29,39,59,35)(32,40,36,41)(33,54,56,58);(1,2)(3,9)(4,10)(5,41,52,11,26,63)(6,54,46,12,25,60)(7,8)(13,28)(14,29)(15,32)(16,33)(17,37,22,34,20,39)(18,62,44,35,50,56)(19,55,49,36,43,64)(21,42,53,38,27,57)(23,58,48,30,47,61)(24,40,45,31,51,59);(2,8)(3,13)(4,15)(5,21)(6,24)(10,33)(11,34)(14,16)(17,23)(20,47)(22,48)(25,51)(26,27)(29,32)(30,38)(35,55)(36,56)(37,41)(39,63)(42,58)(45,46)(52,53)(57,61)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 2, 2]);;
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, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.6*F.8);;
gap> SetConjugate(coll, 8, 4, F.6*F.8);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.6*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);