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#72259758

order2688
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

FittingC2×C7×H(5,𝔽2) ⋊ D3
SylowC7 ⋊ #87516

Subgroups

subgroupG/H
center, FrattiniC2C2×C7⋊#1020
commutator, cosocle, hypocenterC7×#997C22
FittingC2×C7×H(5,𝔽2)D3
socleC2×C7C2×#1020
hypercenterC22C7⋊#1020
LCSG ▷ C7×#997
UCSC1 ◁ C2 ◁ C22

552 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 29 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 — 2 generators on 23 points:

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

Polycyclic presentation:

2^2,3,7,2^5|e^2=i,f^2=i,b^a=bi,c^b=c^2,d^b=d^6,e^b=ei,e^c=f,f^b=ef,f^c=efi,f^e=fi,g^b=h,g^c=h,g^e=gi,h^b=g,h^c=gh,h^e=hi,h^f=hi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72259795
maximal normal subgroup of#72259795

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