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#72342100

order3072
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#11713272
commutator#5950C22×C4
FrattiniC23×C4C22×S4
FittingC2×#21483D3
socleC2#72341577
prime radicalC2×#3025⋊C3C22
cosocleC2×#5950C23
hypercenterC2×C8#1020⋊C2
hypocenter#997C2×C4⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

2246 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 2246 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

100 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, 3, F.5);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 2, 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);;
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);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
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);