Finite Group Databasebeta

← Prev (#72236921) Next (#72236923) →

#72236922

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

Aliases

No aliases for this group.

Splittings

Fitting#60398 ⋊ D3

Subgroups

subgroupG/H
centerC4C2×#1020⋊C2
commutator, cosocle, hypocenter#997C24
FrattiniC23C23×S4
Fitting#60398D3
socleC2C22×#1020⋊C2
hypercenterC2×C4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C4 ◁ C2×C4

3238 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 3238 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:

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

Polycyclic presentation:

2^4,3,2^5|d^2=j,f^2=j,g^2=j,b^a=bj,d^a=dj,e^c=e^2,f^a=fhj,f^c=fg,f^e=g,g^a=ghij,g^c=gj,g^e=fgj,g^f=gj,h^c=i,h^e=hi,h^g=hj,i^c=h,i^e=h,i^f=ij,i^g=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240983, #72240988, #72241002, …
maximal normal subgroup of#72240983, #72240988, #72241002, …
maximal quotient ofC4×#11714182, C4×#11714184, #72341494, …

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