Finite Group Databasebeta

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#72342765

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

Aliases

No aliases for this group.

Splittings

FittingC2×#1791 ⋊ D3

Subgroups

subgroupG/H
centerC4C2×#1020⋊C2
commutator, hypocenter#997C22×C4
FrattiniC22×C4C22×S4
FittingC2×#1791D3
socleC2C2×#1020⋊C4
prime radical#11715827C2
cosocle#5950C23
hypercenterC2×C4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C4 ◁ C2×C4

1385 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 1385 conjugacy classes of subgroups — too many to draw. Only its 49 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:

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

2^4,3,2^5|b^2=d,d^2=j,f^2=j,g^2=j,c^a=cj,c^b=cj,e^c=e^2,f^b=fij,f^c=fj,f^e=g,g^b=ghj,g^c=fg,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

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342771, #72342778, #72342793
maximal normal subgroup of#72342771, #72342778, #72342793

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