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#72342688

order1536
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC2×#3573 ⋊ D3

Subgroups

subgroupG/H
centerC8C2×#1020
commutator, hypocenter#997C22×C4
FrattiniC4C22×#1020
FittingC2×#3573D3
socleC2C2×C4×#1020
prime radical#11715827C2
cosocle#5950C23
hypercenterC2×C8#1020
LCSG ▷ #997
UCSC1 ◁ C8 ◁ C2×C8

2776 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 2776 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^4,3,2^5|a^2=j,b^2=d,d^2=j,f^2=j,g^2=j,c^a=cj,e^c=e^2,f^c=fj,f^e=g,g^c=fg,g^e=fgj,g^f=gj,h^c=hi,h^e=i,h^f=hj,h^g=hj,i^e=hi,i^g=ij

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342708, #72342728, #72342793, …
maximal normal subgroup of#72342708, #72342728, #72342793

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