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#72227698

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C22×B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C25
prime radicalC2×H(7,𝔽2)C22
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

12300 conjugacy classes of subgroups; only the 601 normal subgroups are shown.

normal subgroups only

This group has 12300 conjugacy classes of subgroups — too many to draw. Only its 601 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 — 5 generators on 72 points:

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

Polycyclic presentation:

2^10|b^2=j,c^2=j,f^2=j,g^2=j,b^a=bf,c^a=cg,c^b=cj,d^b=dh,d^c=dh,e^c=eh,f^a=fj,f^b=fj,f^c=fi,f^d=fj,g^a=gj,g^b=gij,g^c=gj,g^d=gj,g^e=gj,g^f=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72227699, #72228046
maximal subgroup of#72227699, #72227738, #72228042
maximal normal subgroup of#72227699, #72227738, #72228042
maximal quotient of#72268023, #72268117, #72268331, …

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