Finite Group Databasebeta

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#72341861

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

Aliases

No aliases for this group.

Splittings

Fitting#392197 ⋊ D3

Subgroups

subgroupG/H
centerC4#11713271
commutator#5950C22×C4
FrattiniC23×C4C22×S4
Fitting#392197D3
socleC2#72341537
prime radical#72341689C2
cosocleC2×#5950C23
hypercenterC2×C8#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

2686 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 2686 conjugacy classes of subgroups — too many to draw. Only its 63 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:

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

Polycyclic presentation:

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

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> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, 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, 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);