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#72277790

order20580
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Splittings

Fitting, HallC5×7−1+2 ⋊ C3×C4
derivedC5×C49 ⋊ C3×C4×C7
Hall#72277779 ⋊ C3
HallC5×(C49⋊C3)⋊C7 ⋊ C4
Hall(C49⋊C3)⋊C7 ⋊ AGL(1,𝔽5)
Sylow7−1+2 ⋊ C3×AGL(1,𝔽5)
SylowC5 ⋊ #72275514

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC5×C49C3×C4×C7
FrattiniC7C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)
FittingC5×7−1+2C3×C4
socleC5×C7C7×ΓL(1,𝔽8)⋊C4
prime radicalD5×(C49⋊C3)⋊C7C2
cosocleD5×C49C2×C3×C7
LCSG ▷ C5×C49
UCSC1

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 245

Polycyclic presentation:

2,3,7,2,5,7^2|a^2=d,f^7=g,e^a=e^2,e^d=e^4,f^a=f^6g^6,f^b=f^4g^2,f^c=fg^3,g^a=g^6,g^b=g^4

Conjugacy classes & character table

137 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#72215772
maximal normal subgroup ofAGL(1,𝔽5)×#72215772

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