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#72272024

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)#453
LCSG ▷ C4×#41⋊C3 ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

1261 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1261 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) = 72
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5)(3,8)(6,7)(9,33,15,53,13,54,32,24)(10,45,20,67,17,68,41,38)(11,36,25,14,23,34,35,31)(12,61,30,51,27,65,58,59)(16,48,39,19,37,46,47,40)(18,70,44,49,26,64,56,69)(21,71,50,55,42,72,66,43)(22,62,52,57,28,63,60,29);(1,2)(3,5)(4,7)(6,8)(12,27)(14,15,36)(16,37)(18,26)(19,20,48)(22,28)(24,25,54)(29,58,63,57,30,62)(31,32,34)(33,53,35)(38,47,68,67,39,45)(40,41,46)(43,56,72,55,44,71)(49,50,70)(51,60,61,59,52,65)(64,69,66);(3,8)(5,6)(9,10)(11,21)(12,26)(13,17)(14,19)(15,20)(16,22)(18,27)(23,42)(24,49)(25,50)(28,37)(29,55)(30,56)(31,40)(32,41)(33,64)(34,46)(35,66)(36,48)(38,51)(39,52)(43,57)(44,58)(45,65)(47,60)(53,69)(54,70)(59,67)(61,68)(62,72)(63,71)

Polycyclic presentation:

2^6,3,2^3|c^2=j,d^2=fj,e^2=j,h^2=j,i^2=j,c^a=ce,c^b=cd,d^a=df,d^b=dfj,d^c=dfj,e^a=ej,e^b=ef,e^c=ej,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.10);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, 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.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*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);