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#72272053

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
Fitting#63001D3
prime radical#11710813C2
hypercenterC4⋊C4#1020
hypocenterQ8⋊SL(2,𝔽3)D4
LCSG ▷ #41⋊SL(2,𝔽3) ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

1039 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 1039 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 384

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

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

Polycyclic presentation:

2^3,3,2^6|a^2=ij,b^2=ij,c^2=i,e^2=j,f^2=j,g^2=i,h^2=i,b^a=bc,c^a=ci,c^b=ci,d^b=d^2,e^b=fj,e^d=f,f^b=ej,f^d=efj,f^e=fj,g^b=gi,g^d=h,g^e=gi,g^f=gi,h^b=gh,h^d=ghi,h^f=hi,h^g=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272057
maximal normal subgroup of#72272057

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