Finite Group Databasebeta

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#72342001

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

2974 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 2974 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342017, #72342055, #72342062
maximal normal subgroup of#72342017, #72342055, #72342062

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.10);;
gap> SetPower(coll, 2, F.4*F.10);;
gap> SetPower(coll, 3, F.4);;
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, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*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, 1, 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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);