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#72266652

order1536
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#128026 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4#1304⋊C3
commutator, cosocle, hypocenter#7317C2×C3
FrattiniC23×Q8C2×A4
Fitting#128026C3
socleC2C2×#1304⋊C3
LCSG ▷ #7317
UCSC1 ◁ C4

517 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 17 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2,3,2^8|a^2=j,c^2=e,d^2=f,e^2=j,f^2=j,g^2=j,c^a=chj,c^b=cd,d^a=dhij,d^b=cefghj,d^c=degj,e^b=fghij,e^d=eh,f^b=eij,f^c=fhi,f^e=fj,g^b=efi,g^c=gh,g^d=gi,g^f=gj,h^b=i,h^d=hj,i^b=hi,i^c=ij,i^d=ij

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72266656
maximal quotient ofC4×#7317⋊C3

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