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#72268253

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#87524
commutator, cosocle#41⋊SL(2,𝔽3)C22
FrattiniC2×C4C2×#1020
FittingQ8×2−1+4D3
prime radical#11710806C2
hypercenterC2×D4#1020
hypocenterQ8⋊SL(2,𝔽3)D4
LCSG ▷ #41⋊SL(2,𝔽3) ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

1059 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^3,3,2^6|a^2=i,c^2=ij,e^2=j,f^2=j,g^2=i,h^2=i,b^a=bc,c^a=cij,c^b=cij,d^a=d^2,e^a=ef,e^d=f,f^a=fj,f^d=efj,f^e=fj,g^a=gh,g^d=h,g^f=gj,h^a=hi,h^d=ghi,h^e=hj,h^g=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268255
maximal normal subgroup of#72268255

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