Finite Group Databasebeta

← Prev (#72226296) Next (#72226298) →

#72226297

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

2841 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 2841 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)(2,43,32,39)(3,88,46,56)(4,5,24,6)(7,33,8,27)(9,25,29,26)(10,76,57,73)(11,48,15,12)(13,44,40,72)(14,45,28,47)(16,52,55,53)(17,18,51,19)(20,58,21,49)(22,61,42,60)(23,64)(34,80,38,35)(36,77,74,95)(37,78,54,79)(41,81)(50,92,75,91)(62,67,66,69)(63,71)(82,89,84,86)(83,87);(1,4,23)(2,50,12,10,22,35)(3,49,41,16,17,81)(5,24,62)(6,25,63)(7,33,66)(8,70,71)(9,27,64)(11,75,39,34,42,73)(13,90,43,36,59,76)(14,91,44,37,60,77)(15,94,72,38,68,95)(18,58,82,20,51,84)(19,88,83,21,52,87)(26,67,29)(28,93,48,54,65,80)(30,31,69)(32,96,45,57,85,78)(40,92,47,74,61,79)(46,53,86,55,56,89);(1,2,3,10,9,11,16,34)(4,22,17,50,27,42,49,75)(5,28,18,54,7,13,20,36)(6,15,19,38,8,32,21,57)(12,41,35,64,39,81,73,23)(14,46,37,29,40,55,74,30)(24,65,51,93,33,59,58,90)(25,68,52,94,70,85,88,96)(26,61,53,92,31,60,56,91)(43,84,76,62,48,82,80,66)(44,86,77,67,47,89,79,69)(45,87,78,63,72,83,95,71)

Polycyclic presentation:

2^4,3^2,2^5|a^2=d,d^2=k,g^2=k,h^2=k,b^a=bdk,c^a=ck,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

70 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, 3, 1, F.3*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);