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#72272035

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#63048D3
prime radical#11710854C2
hypercenterH(3,𝔽4)S4
hypocenterSL(2,𝔽3)#455
LCSG ▷ C4×#41⋊C3 ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

1201 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1201 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272066

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.10);;
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, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
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.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 3, F.9*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);