Finite Group Databasebeta

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#72235035

order1536
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

FittingC22×H(3,𝔽4) ⋊ D3
derivedC22⋊#5464 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22⋊#5464C2
FrattiniC22#87588
FittingC22×H(3,𝔽4)D3
socleC24#1020
LCSG ▷ C22⋊#5464
UCSC1

1529 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 1529 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 28
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,4,3)(5,19,9,8)(6,26,15,24,17,23,14,28)(7,18,10,11)(12,20,22,16,25,21,27,13);(2,3,4)(5,17,25)(6,12,11)(7,15,28)(8,21,22)(9,16,23)(10,20,27)(13,26,18)(14,24,19);(5,7,11,19)(6,13)(8,18,10,9)(12,22,25,27)(14,20)(15,21)(16,17)(23,24,26,28)

Polycyclic presentation:

2,3,2^8|a^2=ij,c^2=j,d^2=i,e^2=ij,f^2=j,b^a=b^2i,c^a=dj,c^b=dj,d^a=ci,d^b=cd,e^a=eij,e^b=fij,e^c=ei,e^d=ej,f^a=ef,f^b=ef,f^c=fj,f^d=fij,g^a=h,g^b=h,h^a=g,h^b=gh,i^a=j,i^b=ij,j^a=i,j^b=i

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235036, #72260914, #72261016, …
maximal normal subgroup of#72235036, #72260914, #72261016, …

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