Finite Group Databasebeta

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#72342726

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4C2×#1020⋊C2
commutator, hypocenter#997C22×C4
FrattiniC22×C4C22×S4
Fitting#60398D3
socleC2C2×#73080
prime radical#60398⋊C3C2
cosocle#5950C23
hypercenterC2×C4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C4 ◁ C2×C4

1211 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 1211 conjugacy classes of subgroups — too many to draw. Only its 65 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:

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

Polycyclic presentation:

2^4,3,2^5|b^2=d,d^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,c^b=cj,e^b=e^2,f^b=fj,f^c=fij,f^e=g,g^b=fg,g^c=ghj,g^e=fgj,g^f=gj,h^b=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, #72342840, …
maximal normal subgroup of#72342728, #72342747, #72342840

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, 2, F.4);;
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, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*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, 2, 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);