Finite Group Databasebeta

← Prev (#72226268) Next (#72226270) →

#72226269

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72215293
commutator, cosocle(#41⋊C3)⋊A4C23
FrattiniC4C2×S42
FittingH(7,𝔽2)D32
hypercenterD4S42
hypocenterΩ+(4,𝔽3)C2×D4
LCSG ▷ (#41⋊C3)⋊A4 ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

2894 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 2894 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

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^a=e^2,e^c=e^2,f^a=e^2f,f^c=f^2,g^a=gk,g^c=gk,g^e=gh,g^f=h,h^a=ghk,h^c=ghk,h^e=g,h^f=gh,h^g=hk,i^a=gi,i^c=ghj,i^e=gik,i^f=j,i^g=ik,i^h=ik,j^a=jk,j^c=hi,j^e=hjk,j^f=ij,j^h=jk

Conjugacy classes & character table

73 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, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.7);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.7*F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 5, F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);