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#72258953

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

3271 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 3271 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)(3,17)(4,60,31,64)(5,28,8,6)(7,29)(12,80,47,81)(13,14,16,39)(18,86,56,89)(19,49,22,20)(21,50)(23,26,57,24)(25,63,27,58)(30,59,61,62)(33,94,74,95)(34,35,37,66)(38,78,44,82)(41,45,77,43)(42,75,79,46)(48,85,53,88)(51,54,84,52)(55,76,83,87)(65,92,71,96)(68,72,91,70)(69,90,93,73);(1,4,23)(2,33,42,10,12,69)(3,48,51,17,18,76)(5,25,63)(6,26,57)(7,30,60)(8,31,64)(9,27,59)(11,65,41,32,38,68)(13,71,81,34,44,95)(14,72,79,35,45,93)(15,73,78,36,46,92)(16,74,82,37,47,96)(19,56,88,22,53,89)(20,83,84,49,54,87)(21,52,86,50,55,85)(24,58,29)(28,61,62)(39,90,77,66,75,91)(40,70,80,67,43,94);(1,2,3,10,9,11,17,32)(4,12,18,33,27,38,48,65)(5,14,50,34,28,15,22,66,29,13,20,67,8,39,21,37,6,40,19,35,7,16,49,36)(23,42,51,69,59,41,76,68)(24,44,54,70,31,75,55,74,26,43,53,72,30,47,83,73,25,45,52,71,61,46,56,90)(57,80,88,93,60,82,87,92,63,79,85,95,62,78,89,91,58,81,84,94,64,77,86,96)

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^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

79 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, 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);