D₁₇₀₈₅ #46803065

order	34170
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₀₈₅
unambiguous semidirect product	C₃⋊D₅₆₉₅
unambiguous semidirect product	C₅⋊D₃₄₁₇
unambiguous semidirect product	C₁₇⋊D₁₀₀₅
unambiguous semidirect product	C₆₇⋊D₂₅₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₇×C₆₇ ⋊ C₂
Hall	C₅×C₁₇×C₆₇ ⋊ D₃
Hall	C₃×C₁₇×C₆₇ ⋊ D₅
Hall	C₁₇×C₆₇ ⋊ D₁₅
Hall	C₃×C₅×C₆₇ ⋊ D₁₇
Hall	C₅×C₆₇ ⋊ D₅₁
Hall	C₃×C₅×C₁₇ ⋊ D₆₇
Hall	C₃×C₆₇ ⋊ D₈₅
Hall	C₅×C₁₇ ⋊ D₂₀₁
Sylow	C₆₇ ⋊ D₂₅₅
Hall	C₃×C₁₇ ⋊ D₃₃₅
Sylow	C₁₇ ⋊ D₁₀₀₅
Hall	C₃×C₅ ⋊ D₁₁₃₉
Sylow	C₅ ⋊ D₃₄₁₇
Sylow	C₃ ⋊ D₅₆₉₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₇×C₆₇	C₂
LCS	G ▷ C₃×C₅×C₁₇×C₆₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 17085

A minimal faithful permutation representation — 2 generators on 92 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92);(2,3)(5,8)(6,7)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2,3,5,17,67|b^a=b^2,c^a=c^4,d^a=d^16,e^a=e^66

Conjugacy classes & character table

8544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34170, 17);

D17085 #46803065