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#72246255

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

Aliases

No aliases for this group.

Splittings

FittingC7×H(5,𝔽2) ⋊ C3⋊D3
derivedC7×Ω+(4,𝔽3) ⋊ C2
SylowC7 ⋊ SO+(4,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#72222237
commutator, cosocle, hypocenterC7×Ω+(4,𝔽3)C2
FittingC7×H(5,𝔽2)C3⋊D3
socleC2×C7A4⋊S4
LCSG ▷ C7×Ω+(4,𝔽3)
UCSC1 ◁ C2

254 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 25 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 — 3 generators on 31 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72246260, D7×SO+(4,𝔽3)
maximal normal subgroup of#72246260, D7×SO+(4,𝔽3)

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