Finite Group Databasebeta

← Prev (#72226325) Next (#72226327) →

#72226326

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

Aliases

No aliases for this group.

Splittings

Fitting#585 ⋊ D32

Subgroups

subgroupG/H
center, hypercenterC4S42
commutator, cosocle, hypocenterΩ+(4,𝔽3)C23
Frattini, socleC2C2×S42
Fitting#585D32
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ 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) = 64
transitive degreeμt(G) = 64

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

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

Polycyclic presentation:

2^3,3^2,2^5|a^2=j,c^2=j,f^2=j,g^2=j,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

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228062
maximal normal subgroup of#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, 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);