#72244734

order	3400
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅²×C₁₇ ⋊ C₈
Sylow	C₅² ⋊ #3633
Sylow	C₁₇ ⋊ #6033

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₇	C₈
prime radical	C₅²×D₁₇	C₄
cosocle	#13133353	C₂
LCS	G ▷ C₅²×C₁₇
UCS	C₁

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 425

Polycyclic presentation:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72244735, #72244799
maximal normal subgroup of	#72244735, #72244799

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 5, 17]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^3);;
gap> SetConjugate(coll, 4, 2, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^3);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^4);;
gap> SetConjugate(coll, 6, 3, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);