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#72226311

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

Aliases

No aliases for this group.

Splittings

FittingH(7,𝔽2) ⋊ D32

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

2836 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 2836 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:

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

Polycyclic presentation:

2^4,3^2,2^5|a^2=d,d^2=k,g^2=k,h^2=k,b^a=bd,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);;
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);