Finite Group Databasebeta

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#72341449

order1536
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#20608 ⋊ D3

Subgroups

subgroupG/H
centerC4#72991
commutator#5950C2×C4
FrattiniC23×C4C2×S4
Fitting#20608D3
socleC2#11711331
prime radical#11715827C2
cosocleC2×#5950C22
hypercenterC2×C4#1020⋊C2
hypocenter#997C22⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4

1138 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 29 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 — 2 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341458, #72341463, #72341508
maximal normal subgroup of#72341458, #72341463, #72341508

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