#72246902

order	7800
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅²×C₁₃ ⋊ C₂×C₃×C₄
Hall	#72246844 ⋊ C₃
Hall	C₅²×C₁₃⋊C₃ ⋊ C₂×C₄
Hall	C₁₃⋊C₃ ⋊ #6036
Sylow	C₅² ⋊ C₂×AGL(1,𝔽₁₃)
Sylow	C₁₃ ⋊ C₃×#6036

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₃	C₂×C₃×C₄
prime radical	C₅⋊D₅×D₁₃⋊C₃	C₂
cosocle	C₅²×D₁₃	C₂²×C₃
LCS	G ▷ C₅²×C₁₃
UCS	C₁

168 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 130

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72246910, D₅²×AGL(1,𝔽₁₃), #72246954
maximal normal subgroup of	#72246910, D₅²×AGL(1,𝔽₁₃), #72246954

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