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#72246258

order4032
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

FittingC7×H(5,𝔽2) ⋊ C3×D3
SylowC7 ⋊ C3×#5429

Subgroups

subgroupG/H
centerC2#72222240
commutator, cosocleC7×#41⋊C3C22×C3
FrattiniC4C2×A4⋊AGL(1,𝔽7)
FittingC7×H(5,𝔽2)C3×D3
socleC2×C7C3×GL(2,ℤ/4)
hypercenterQ8A4⋊AGL(1,𝔽7)
hypocenterC7×SL(2,𝔽3)C3×D4
LCSG ▷ C7×#41⋊C3 ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

334 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 31
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 2 generators on 31 points:

(1,17,19,2,11,22,7,16,24,6,14,12)(3,9,23,5,15,21,10,13,18,8,4,20)(25,26,31)(28,29,30);(2,5)(3,10)(4,12)(6,8)(9,18)(11,20)(13,23)(14,21)(15,22)(16,19)(17,24)(26,27)(28,29)(30,31)

Polycyclic presentation:

2^2,3,2,3,7,2^3|b^2=i,d^2=i,g^2=i,h^2=i,b^a=bd,d^a=di,d^b=di,e^a=e^2,f^a=f^6,f^c=f^2,g^a=hi,g^e=gh,h^a=gi,h^e=g,h^g=hi

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72246260, AGL(1,𝔽7)×#5429
maximal normal subgroup ofAGL(1,𝔽7)×#5429

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