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#72341996

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#6629044 ⋊ D3

Subgroups

subgroupG/H
center, socleC2C2×#11711262
commutator, cosocle#85666C23
FrattiniC22×C8C22×S4
Fitting#6629044D3
hypercenterC2×C8#1020⋊C2
hypocenter#997C2×D8
LCSG ▷ #85666 ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

2362 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 2362 conjugacy classes of subgroups — too many to draw. Only its 53 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:

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

Polycyclic presentation:

2^5,3,2^5|b^2=de,d^2=ek,e^2=k,g^2=k,h^2=k,b^a=bd,c^a=ck,c^b=ck,d^a=de,e^a=ek,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

70 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*F.5);;
gap> SetPower(coll, 4, F.5*F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*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);