Finite Group Databasebeta

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#72342024

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingD4×H(5,𝔽2) ⋊ D3

Subgroups

subgroupG/H
center, socleC22#87524
commutator, cosocleC4×#997C22
FrattiniC2×C4C2×#1020
FittingD4×H(5,𝔽2)D3
hypercenterC2×D4#1020
hypocenter#997D8
LCSG ▷ C4×#997 ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

1937 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 1937 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 16
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342031, #72342067, #72342082
maximal normal subgroup of#72342031, #72342067, #72342082

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