#72244792

order	6800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅²×C₁₇ ⋊ C₂²×C₄
Sylow	C₅² ⋊ C₂²×#595
Sylow	C₁₇ ⋊ C₂×D₅⋊AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₇	C₂²×C₄
prime radical	D₅²×D₁₇	C₂
cosocle	C₅⋊D₈₅	C₂³
LCS	G ▷ C₅²×C₁₇
UCS	C₁

248 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 170

Polycyclic presentation:

2^4,5^2,17|b^2=d,e^a=e^4,e^b=e^3,e^d=e^4,f^a=e^3f,f^b=f^3,f^d=f^4,g^b=g^13,g^c=g^16,g^d=g^16

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#595×D₅⋊AGL(1,𝔽₅)
maximal normal subgroup of	#595×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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 5, 17]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 5, 2, F.5^3);;
gap> SetConjugate(coll, 5, 4, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.5^3*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 2, F.7^13);;
gap> SetConjugate(coll, 7, 3, F.7^16);;
gap> SetConjugate(coll, 7, 4, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);