Finite Group Databasebeta

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#72236919

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

Aliases

No aliases for this group.

Splittings

Fitting#32896 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713200
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#32896D3
hypercenterD4#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ D4

1401 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 1401 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:

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

Polycyclic presentation:

2^4,3,2^5|b^2=d,c^2=dj,d^2=j,f^2=j,g^2=j,b^a=bd,c^a=cd,c^b=cj,d^a=dj,e^c=e^2,f^b=fhij,f^c=fgij,f^e=g,g^b=ghj,g^c=gh,g^e=fgj,g^f=gj,h^e=i,h^f=hj,i^c=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#72240981, #72240996, #72241002, …
maximal normal subgroup of#72240981, #72240996, #72241002, …
maximal quotient of#72341898, C2×#72236919, #72343706

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