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#72236829

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

Aliases

No aliases for this group.

Splittings

Fitting#21484 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713272
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#21484D3
hypercenterC8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ C8

952 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 39 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237002, #72240981, #72240986, …
maximal normal subgroup of#72237002, #72240981, #72240986, …
maximal quotient of#72341891, C2×#72236829, #72343907

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