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#72228369

order2304
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingH(7,𝔽2) ⋊ C3×D3

Subgroups

subgroupG/H
center, Frattini, socleC2C22×A4×S4
commutator, cosocle, hypocenter#994C23×C3
FittingH(7,𝔽2)C3×D3
hypercenterD4A4×S4
LCSG ▷ #994
UCSC1 ◁ C2 ◁ D4

2246 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 2246 conjugacy classes of subgroups — too many to draw. Only its 79 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:

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

Polycyclic presentation:

2^3,3,2^2,3,2^3|e^2=j,f^2=j,h^2=j,i^2=j,b^a=bj,c^a=cj,c^b=cj,e^d=f,f^d=efj,f^e=fj,g^a=g^2,h^a=hj,h^g=i,i^a=hi,i^g=hij,i^h=ij

Conjugacy classes & character table

89 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 3, 2, 2, 2]);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 4, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);