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#72258356

order1536
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#13794 ⋊ D3
derived#13794⋊C3 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC22#85547
commutator, cosocle, hypocenter#13794⋊C3C2
Frattini, socleC24#1020
Fitting#13794D3
LCSG ▷ #13794⋊C3
UCSC1 ◁ C22

1236 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 1236 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) = 48

A minimal faithful permutation representation — 2 generators on 28 points:

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

Polycyclic presentation:

2,3,2^8|c^2=h,d^2=hi,b^a=b^2,c^a=dh,c^b=dhj,d^a=chi,d^b=cdj,d^c=dj,e^b=f,e^c=eg,e^d=eg,f^a=ef,f^b=ef,f^d=fg,h^a=hi,h^b=hi,i^b=hj

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258367
maximal normal subgroup of#72258367

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