Finite Group Databasebeta

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#72341580

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

Aliases

No aliases for this group.

Splittings

Fitting#20610 ⋊ D3

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, cosocleC2×#997C23
FrattiniC24C22×S4
Fitting#20610D3
hypercenterC2×C4#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4

1860 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1860 conjugacy classes of subgroups — too many to draw. Only its 43 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341587, #72341800, #72341827, …
maximal normal subgroup of#72341587, #72341800, #72341827, …

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