Finite Group Databasebeta

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#72236895

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

Aliases

No aliases for this group.

Splittings

Fitting#32894 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713200
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#32894D3
hypercenterD4#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ D4

1468 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 1468 conjugacy classes of subgroups — too many to draw. Only its 35 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:

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

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^a=e^2,e^b=e^2,f^a=fgij,f^b=fgij,f^c=fhij,f^e=g,g^a=gh,g^b=gh,g^c=ghj,g^e=fgj,g^f=gj,h^e=i,h^f=hj,i^a=hi,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, #72240992, #72241001, …
maximal normal subgroup of#72237002, #72240992, #72241001, …
maximal quotient of#72342029, #72342056, C2×#72236895, …

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