Finite Group Databasebeta

← Prev (#72258940) Next (#72258942) →

#72258941

order4608
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24×C3

Aliases

No aliases for this group.

Splittings

Fitting#63092 ⋊ C3⋊D3

Subgroups

subgroupG/H
center, socleC2D4×A4⋊S4
commutator, cosocle(#41⋊C3)⋊A4C23
FrattiniC4C22×A4⋊S4
Fitting#63092C3⋊D3
hypercenterSD16A4⋊S4
hypocenterΩ+(4,𝔽3)C2×D4
LCSG ▷ (#41⋊C3)⋊A4 ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ SD16

3258 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 3258 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 96
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 96 points:

(1,9)(2,16)(3,17)(4,59,31,63)(5,28,8,6)(7,29)(10,37)(11,76,46,80)(12,43,15,13)(14,44)(18,85,55,88)(19,48,22,20)(21,49)(23,26,56,24)(25,62,27,57)(30,58,60,61)(32,92,72,96)(33,69,36,34)(35,70)(38,41,73,39)(40,79,42,74)(45,75,77,78)(47,84,52,87)(50,53,83,51)(54,81,82,86)(64,67,89,65)(66,95,68,90)(71,91,93,94);(1,4,23)(2,32,38,10,11,64)(3,47,50,17,18,81)(5,25,62)(6,26,56)(7,30,59)(8,31,63)(9,27,58)(12,66,79,33,40,95)(13,67,73,34,41,89)(14,71,76,35,45,92)(15,72,80,36,46,96)(16,68,75,37,42,91)(19,55,87,22,52,88)(20,82,83,48,53,86)(21,51,85,49,54,84)(24,57,29)(28,60,61)(39,90,44,65,74,70)(43,93,78,69,77,94);(1,2,3,10,9,16,17,37)(4,11,18,32,27,42,47,68)(5,13,49,33,28,14,22,69,29,12,20,70,8,43,21,36,6,44,19,34,7,15,48,35)(23,38,50,64,58,75,81,91)(24,40,53,65,31,77,54,72,26,39,52,67,30,46,82,71,25,41,51,66,60,45,55,93)(56,74,87,89,59,80,86,92,62,73,84,95,61,76,88,94,57,79,83,90,63,78,85,96)

Polycyclic presentation:

2^4,3^2,2^5|a^2=d,d^2=k,g^2=k,h^2=k,b^a=bdk,d^b=dk,e^c=e^2,f^c=f^2,g^c=hk,g^e=h,g^f=h,h^c=gk,h^e=ghk,h^f=ghk,h^g=hk,i^c=gij,i^e=j,i^f=ghi,i^h=ik,j^c=ghjk,j^e=ij,j^f=gjk,j^g=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, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 3, F.8*F.11);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 5, F.7*F.8*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, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 6, F.7*F.8*F.9);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> SetConjugate(coll, 10, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);