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#72341913

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#11713271
commutator#5950C22×C4
FrattiniC23×C4C22×S4
Fitting#6629044D3
socleC2#72341540
prime radical#72341679C2
cosocleC2×#5950C23
hypercenterC2×C8#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

2318 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 2318 conjugacy classes of subgroups — too many to draw. Only its 79 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:

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

Conjugacy classes & character table

82 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, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 2, F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
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);