Finite Group Databasebeta

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#72342738

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

Aliases

No aliases for this group.

Splittings

Fitting#21483 ⋊ D3

Subgroups

subgroupG/H
centerC4C2×#1020⋊C2
commutator, hypocenter#997C22×C4
FrattiniC22×C4C22×S4
Fitting#21483D3
socleC2C4×#1020⋊C2
prime radical#11714169C2
cosocle#5950C23
hypercenterC8#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C4 ◁ C8

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

normal subgroups only

This group has 914 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:

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

Polycyclic presentation:

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

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342747, #72342771, #72342793
maximal normal subgroup of#72342747, #72342771, #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, 1, F.10);;
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, 1, 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, 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);