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#72230265

order2688
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC7×#585 ⋊ C2×C3
Hall#11744566 ⋊ C3
SylowC7 ⋊ #87547

Subgroups

subgroupG/H
center, FrattiniC2C2×D7⋊(C22⋊A4)
commutator, cosocle, hypocenterC7×H(5,𝔽2)C22×C3
FittingC7×#585C2×C3
socleC2×C7C22×C22⋊A4
hypercenterC4D7⋊(C22⋊A4)
LCSG ▷ C7×H(5,𝔽2)
UCSC1 ◁ C2 ◁ C4

2154 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 2154 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(2,6)(3,9,4)(7,16,8,13,15,14)(10,11,12)(18,21,22,19,20,23);(1,2,5,6)(3,13,10,7)(4,14,12,8)(9,15,11,16)(17,18,20,23,22,21,19);(1,11,5,9)(2,16,6,15)(3,4,10,12)(7,8,13,14)

Polycyclic presentation:

2^2,3,2^4,7,2|a^2=i,d^2=i,e^2=i,b^a=bi,d^c=e,e^c=dei,e^d=ei,f^c=fg,f^e=fi,g^c=f,g^d=gi,g^e=gi,h^b=h^6,h^c=h^2

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72230283
maximal normal subgroup of#72230283

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