Finite Group Databasebeta

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#72246944

order15600
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, HallC52×C13 ⋊ C22×C3×C4
Hall#72246865 ⋊ C3
HallC52×C13⋊C3 ⋊ C22×C4
HallC13⋊C3 ⋊ C2×D5⋊AGL(1,𝔽5)
SylowC52 ⋊ C22×AGL(1,𝔽13)
SylowC13 ⋊ C2×C3×D5⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC52×C13C22×C3×C4
prime radicalD52×D13⋊C3C2
cosocleC5⋊D65C23×C3
LCSG ▷ C52×C13
UCSC1

496 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 130

Polycyclic presentation:

2^3,3,2,5^2,13|b^2=e,f^a=f^4,f^b=f^3,f^c=f^4,f^e=f^4,g^a=f^3g,g^b=g^3,g^c=g^4,g^e=g^4,h^b=h^8,h^d=h^3,h^e=h^12

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽13)×D5⋊AGL(1,𝔽5)
maximal normal subgroup ofAGL(1,𝔽13)×D5⋊AGL(1,𝔽5)

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 5, 5, 13]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 6, 5, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^3*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 7, 5, F.7^4);;
gap> SetConjugate(coll, 8, 2, F.8^8);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> SetConjugate(coll, 8, 5, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);