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#72238834

order1536
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#7633414 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#11710658
commutator, Fitting, cosocle, hypocenter#7633414C3
Frattini, socleC23C22⋊(C22⋊A4)
LCSG ▷ #7633414
UCSC1 ◁ C2

1607 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 1607 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7)(3,13,36)(4,17,39)(5,10,27)(6,12,26)(8,25,18)(9,31,15)(11,34,23)(14,29,43)(16,32,47)(19,28,24)(20,40,45)(21,35,48)(22,41,42)(30,44,37)(33,46,38);(1,23,6,5)(3,16,15,21)(4,22,19,20)(7,27,26,34)(8,29)(9,13)(14,37,18,38)(24,42,39,45)(25,46,43,44)(30,33)(31,48,36,47)(32,35);(4,19)(8,29)(9,13)(14,18)(20,22)(24,39)(30,33)(31,36)(32,35)(37,38)(42,45)(47,48);(1,18)(2,28,12,17)(3,19)(4,15)(5,38)(6,14)(7,31)(8,9,29,13)(10,40,11,41)(16,22)(20,21)(23,37)(24,25)(26,36)(27,47)(30,35,33,32)(34,48)(39,43)(42,44)(45,46)

Polycyclic presentation:

3,2^9|b^2=h,c^2=hj,f^2=i,g^2=ij,b^a=bc,c^a=bij,c^b=chi,d^a=de,d^b=di,d^c=dj,e^a=d,e^b=ej,e^c=eij,f^a=fg,f^b=fi,f^c=fj,g^a=fij,g^b=gj,g^c=gij,h^a=hij,i^a=j,j^a=ij

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238835
maximal normal subgroup of#72238835

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