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#72228100

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#63092 ⋊ C32

Subgroups

subgroupG/H
center, socleC2D4×A42
commutator, cosocle#585C22×C32
FrattiniC4C22×A42
Fitting#63092C32
hypercenterSD16A42
hypocenterH(5,𝔽2)C32×D4
LCSG ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C2 ◁ C4 ◁ SD16

1070 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 1070 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:

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

Polycyclic presentation:

2^2,3^2,2^6|a^2=e,e^2=j,f^2=j,g^2=j,b^a=bej,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, #72258941, #72258953
maximal normal subgroup of#72258914, #72258941, #72258953
maximal quotient of#72253052

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*F.10);;
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);