Finite Group Databasebeta

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#72342718

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

Aliases

No aliases for this group.

Splittings

Fitting#21484 ⋊ D3

Subgroups

subgroupG/H
centerC4C2×#1020⋊C2
commutator, hypocenter#997C22×C4
FrattiniC22×C4C22×S4
Fitting#21484D3
socleC2C4×#1020⋊C2
prime radical#11714169C2
cosocle#5950C23
hypercenterC8#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C4 ◁ C8

918 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 47 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:

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

Polycyclic presentation:

2^4,3,2^5|a^2=j,b^2=d,c^2=j,d^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,c^b=cj,e^c=e^2,f^a=fij,f^b=fij,f^c=fi,f^e=g,g^a=ghj,g^b=ghj,g^c=fghij,g^e=fgj,g^f=gj,h^c=hi,h^e=hi,h^f=hj,i^e=h,i^g=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342728, #72342747, #72342800
maximal normal subgroup of#72342728, #72342747, #72342800

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