Finite Group Databasebeta

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#72342743

order1536
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#6372743 ⋊ C3

Subgroups

subgroupG/H
centerC4C22×#120⋊C3
commutator, hypocenterH(5,𝔽2)C22×C3×C4
FrattiniC22×C4C23×A4
Fitting#6372743C3
socleC2C2×C4×#120⋊C3
prime radical#60398⋊C3C2
cosocle#585C23×C3
hypercenterM16#120⋊C3
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C4 ◁ M16

1654 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1654 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^3,3,2^6|b^2=e,e^2=j,f^2=j,g^2=j,c^a=cj,c^b=cj,f^b=fi,f^c=fi,f^d=g,g^b=gh,g^c=gh,g^d=fgj,g^f=gj,h^d=hij,h^f=hj,i^d=h,i^g=ij

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342747, #72342771, #72342826
maximal normal subgroup of#72342747, #72342771, #72342826

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