Finite Group Databasebeta

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#72344071

order1536
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

Fitting#60401 ⋊ D3

Subgroups

subgroupG/H
center, socleC2C22×#1020⋊C2
commutator, cosocle, hypocenter#997C24
FrattiniC23C23×S4
Fitting#60401D3
hypercenterD4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C2 ◁ D4

3313 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 3313 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,18,17,30,6,12,5,29)(2,27,22,32,7,23,10,31)(3,4,15,16,11,14,13,19)(8,9,25,26,20,21,24,28);(2,7)(4,5)(8,20)(9,22)(10,21)(12,13)(14,17)(15,18)(23,25)(24,27)(26,28)(31,32);(1,2,6,7)(3,8,11,20)(4,9,14,21)(5,10,17,22)(12,23,18,27)(13,24,15,25)(16,26,19,28)(29,31,30,32);(3,11)(4,14)(5,17)(8,20)(9,21)(10,22)(16,19)(26,28)

Polycyclic presentation:

2^4,3,2^5|b^2=j,d^2=j,f^2=j,g^2=j,c^b=cj,d^a=dj,e^c=e^2,e^d=e^2,f^a=fhij,f^c=gh,f^d=gj,f^e=g,g^a=ghj,g^c=fhi,g^d=fj,g^e=fgj,g^f=gj,h^c=hi,h^d=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72344080
maximal normal subgroup of#72344080
maximal quotient of#72344095, #72344117

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