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#72228293

order2304
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

FittingH(7,𝔽2) ⋊ C3⋊D3

Subgroups

subgroupG/H
center, socleC2#11923523
commutator, cosocle(#41⋊C3)⋊A4C22
FrattiniC4C2×A4⋊S4
FittingH(7,𝔽2)C3⋊D3
hypercenterD4A4⋊S4
hypocenterΩ+(4,𝔽3)D4
LCSG ▷ (#41⋊C3)⋊A4 ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

854 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 37 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:

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

Polycyclic presentation:

2^3,3^2,2^5|a^2=c,c^2=j,f^2=j,g^2=j,b^a=bc,c^b=cj,d^a=d^2,e^a=e^2,f^a=gj,f^d=g,f^e=g,g^a=fj,g^d=fgj,g^e=fgj,g^f=gj,h^a=fhi,h^d=i,h^e=fgh,h^g=hj,i^a=fgij,i^d=hi,i^e=fij,i^f=ij

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258953, #72258961
maximal normal subgroup of#72258953, #72258961

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 5, F.6*F.7*F.8);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);