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#72260656

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#35924 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#11715691
commutator, cosocle, hypocenter#2273⋊C3C22
Frattini, socleC23C2×#1020
Fitting#35924D3
LCSG ▷ #2273⋊C3
UCSC1 ◁ C2

697 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 40
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 40 points:

(1,10,18,2,11,23)(3,24,4,6,22,7)(5,16,9,14,21,19)(8,12,15,13,20,17)(25,38,35,32,34,40,29,30,36,26,27,39)(28,31,33,37);(2,7)(3,9,11,17)(6,19)(8,14)(10,15)(12,22,21,18)(16,23)(20,24)(27,28,34,33)(30,31,38,37)

Polycyclic presentation:

2^2,3,2^7|a^2=hi,b^2=j,f^2=i,g^2=h,b^a=bhi,c^a=c^2hj,c^b=chi,d^a=deij,d^b=dhj,d^c=e,e^a=ehi,e^b=ehi,e^c=de,f^a=gi,f^b=fij,f^c=gij,f^d=fj,f^e=fj,g^a=fh,g^b=ghj,g^c=fgj,g^e=gj,g^f=gj,h^a=i,h^c=hij,i^a=h,i^c=h

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72260665
maximal normal subgroup of#72260665

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