Finite Group Databasebeta

← Prev (#72228100) Next (#72228102) →

#72228101

order2304
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#63091 ⋊ C32

Subgroups

subgroupG/H
center, socleC2D4×A42
commutator, cosocle#585C22×C32
FrattiniC4C22×A42
Fitting#63091C32
hypercenterD8A42
hypocenterH(5,𝔽2)C32×D4
LCSG ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C2 ◁ C4 ◁ D8

1078 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 67 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 — 2 generators on 96 points:

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

Polycyclic presentation:

2^2,3^2,2^6|a^2=e,e^2=j,f^2=j,g^2=j,b^a=be,e^b=ej,f^c=fg,f^d=fg,g^c=f,g^d=f,g^f=gj,h^c=i,h^d=fhj,h^f=hj,h^g=hj,i^c=hi,i^d=fgij,i^f=ij

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258914, #72258961, C2×#72228101
maximal normal subgroup of#72258914, #72258961, C2×#72228101
maximal quotient ofC2×#72228101

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