Finite Group Databasebeta

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#72344109

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

Aliases

No aliases for this group.

Splittings

FittingC2×#3026 ⋊ D3

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, cosocleC2×#997C23
FrattiniC24C22×S4
FittingC2×#3026D3
hypercenterC23#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C23

1648 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1648 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 32

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

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

Polycyclic presentation:

2^4,3,2^5|b^2=d,f^2=j,g^2=j,b^a=bd,c^a=cj,c^b=cj,e^b=e^2,f^b=gh,f^c=fhij,f^e=g,g^b=fhi,g^c=ghj,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

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72344110, #72344113, #72344117
maximal normal subgroup of#72344110, #72344113, #72344117

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