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#72227896

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2×C4#73027
commutator, cosocle#52291⋊C3C22
FrattiniC22×#41C2×S4
Fitting#498989D3
socleC22#11711626
prime radical#72227888C2
hypercenterC22×C4#1020⋊C2
hypocenter#73291D4
LCSG ▷ #52291⋊C3 ▷ #73291
UCSC1 ◁ C2×C4 ◁ C22×C4

1005 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 1005 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 128

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^3,3,2^7|b^2=jk,c^2=k,e^2=i,f^2=i,g^2=k,h^2=k,b^a=bc,c^a=ck,c^b=ck,d^b=d^2,e^a=eghijk,e^b=efhijk,e^d=f,f^a=fgij,f^b=fgjk,f^d=efi,f^e=fi,g^b=gk,g^d=h,g^e=gij,h^b=gh,h^d=ghk,h^e=hij,h^f=hij,h^g=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227900

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