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#72272121

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#85546
commutator, cosocleC4×#41⋊C3C23
FrattiniC42C22×S4
Fitting#63016D3
prime radical#63016⋊C3C2
hypercenterD42S4
hypocenterSL(2,𝔽3)#455
LCSG ▷ C4×#41⋊C3 ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ D42

1311 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 1311 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^6,3,2^3|c^2=fj,d^2=f,e^2=fj,h^2=j,i^2=j,c^a=ce,c^b=cd,d^b=df,d^c=df,e^a=efj,e^c=efj,g^c=g^2,h^c=ij,h^g=i,i^c=hj,i^g=hij,i^h=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272131

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