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#72341899

order3072
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#10574037 ⋊ D3

Subgroups

subgroupG/H
centerC4C2×C4⋊#1020
commutator#5950C22×C4
FrattiniC2×C4C22×#1020
Fitting#10574037D3
socleC2C4×C4⋊#1020
prime radical#72341689C2
cosocleC2×#5950C23
hypercenter#97#1020
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C8 ◁ #97

5750 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 5750 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^5,3,2^5|a^2=d,b^2=de,c^2=e,d^2=k,e^2=k,b^a=be,c^a=ce,c^b=ck,e^a=ek,f^a=f^2,g^a=h,g^f=h,h^a=g,h^f=gh,i^a=ij,i^f=j,i^g=ik,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

118 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, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.5);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, 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);