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#72342051

order3072
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC8C2×#1020⋊C2
commutator, hypocenter#997C22×C8
FrattiniC22×C8C22×S4
FittingC2×#21483D3
socleC2C2×#11711113
prime radicalC2×#3025⋊C3C4
cosocle#85666C23
hypercenterC2×C8#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C8 ◁ C2×C8

1822 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 1822 conjugacy classes of subgroups — too many to draw. Only its 101 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:

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

Polycyclic presentation:

2^5,3,2^5|b^2=d,d^2=e,e^2=k,g^2=k,h^2=k,c^b=ck,f^b=f^2,g^a=gijk,g^b=hk,g^f=h,h^a=hik,h^b=gk,h^f=ghk,h^g=hk,i^b=ij,i^f=j,i^g=ik,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.8*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);