Finite Group Databasebeta

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#72259795

order8064
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

FittingC2×C7×H(5,𝔽2) ⋊ C3×D3
SylowC7 ⋊ C3×#87516

Subgroups

subgroupG/H
center, FrattiniC2C2×#72214032
commutator, cosocle, hypocenterC7×#997C22×C3
FittingC2×C7×H(5,𝔽2)C3×D3
socleC2×C7C2×C3×#1020
hypercenterC22#72214032
LCSG ▷ C7×#997
UCSC1 ◁ C2 ◁ C22

1156 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 45 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:

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

Polycyclic presentation:

2^2,3^2,7,2^5|f^2=j,g^2=j,b^a=bj,d^b=d^2,e^b=e^6,e^c=e^2,f^b=fj,f^d=g,g^b=fg,g^d=fgj,g^f=gj,h^b=i,h^d=i,h^f=hj,i^b=h,i^d=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

91 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, 3, 3, 7, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^6);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);