Finite Group Databasebeta

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#72236799

order1536
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#6573736 ⋊ C3

Subgroups

subgroupG/H
center, socleC2D4×#120⋊C3
commutator, cosocle#585C23×C3
FrattiniC22×C4C23×A4
Fitting#6573736C3
hypercenterSD16#120⋊C3
hypocenterH(5,𝔽2)C2×C3×D4
LCSG ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C2 ◁ C4 ◁ SD16

1691 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 1691 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^3,3,2^6|c^2=e,e^2=j,f^2=j,g^2=j,c^a=cj,c^b=ce,e^b=ej,f^a=fhi,f^b=fhi,f^c=fhi,f^d=g,g^a=gh,g^b=gh,g^c=gh,g^d=fgj,g^f=gj,h^d=ij,h^f=hj,i^d=hij,i^f=ij,i^g=ij

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237002, #72240977, #72240982, …
maximal normal subgroup of#72237002, #72240977, #72240982, …
maximal quotient of#72341828, C2×#72236799, #72343896

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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, 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.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.8*F.9*F.10);;
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);