Finite Group Databasebeta

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#72341855

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

Aliases

No aliases for this group.

Splittings

Fitting#20612 ⋊ D3

Subgroups

subgroupG/H
centerC4#73081
commutator#5950C2×C4
FrattiniC23×C4C2×S4
Fitting#20612D3
socleC2#11711331
prime radical#11715838C2
cosocleC2×#5950C22
hypercenterC2×C4#1020⋊C2
hypocenter#997C22⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4

1161 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 1161 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341861
maximal normal subgroup of#72341861

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