Finite Group Databasebeta

← Prev (#72342798) Next (#72342800) →

#72342799

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

Aliases

No aliases for this group.

Splittings

FittingC22×#508 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#73133
commutator, cosocleA4×C16C23
FrattiniC16C22×S4
FittingC22×#508D3
socleC23C2×#826
hypercenter#508S4
hypocenterA4#2263
LCSG ▷ A4×C16 ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #508

576 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 36
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^7,3,2^2|a^2=d,d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,c^b=cg,d^b=de,e^b=ef,f^b=fg,h^a=h^2,i^a=ij,i^h=j,j^h=ij

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342806, S4×#2263, #72342908
maximal normal subgroup of#72342806, S4×#2263, #72342908

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