Finite Group Databasebeta

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#72344100

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC4×#585 ⋊ D3

Subgroups

subgroupG/H
centerC2×C4C2×#1020
commutator, cosocleC2×#997C23
Frattini, socleC22C22×#1020
FittingC4×#585D3
hypercenterC42#1020
hypocenter#997C2×D4
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C2×C4 ◁ C42

3241 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 3241 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,3,4)(7,16,15)(8,13,14)(10,20,19)(11,17,18);(5,18,9,10)(6,7,12,14)(8,19,16,17)(11,13,20,15);(2,4)(7,8)(10,11)(14,16)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72344102, #72344121, #72344125
maximal normal subgroup of#72344102, #72344121, #72344125

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