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#72218335

order6720
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

FittingC5×C7×Q8 ⋊ C2×C3×C4
Hall#72218334 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ #417
HallC5×C7 ⋊ C3×#417
HallΓL(1,𝔽8) ⋊ #65793
SylowC7 ⋊ C3×#65793
SylowC5 ⋊ #89575⋊C3

Subgroups

subgroupG/H
centerC2AGL(1,𝔽7)×C22⋊AGL(1,𝔽5)
commutatorC4×C5×C7C22×C3×C4
FrattiniC4C2×AGL(1,𝔽5)×AGL(1,𝔽7)
FittingC5×C7×Q8C2×C3×C4
socleC2×C5×C7C2×C3×C22⋊C4
prime radicalD5×#66659C2
cosocleC4×C7×D5C23×C3
hypercenterQ8AGL(1,𝔽5)×AGL(1,𝔽7)
hypocenterC5×C7C3×#417
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ Q8

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^3,5,7|a^2=e,b^2=efg,e^2=g,f^2=g,b^a=bf,c^a=cg,e^b=eg,f^b=fg,h^a=h^3,h^b=h^3,h^e=h^4,i^b=i^6,i^c=i^6,i^d=i^2

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#6477⋊C3, #72218566, AGL(1,𝔽7)×#65793
maximal normal subgroup ofAGL(1,𝔽5)×#6477⋊C3, #72218566, AGL(1,𝔽7)×#65793

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