Finite Group Databasebeta

← Prev (#145) Next (#147) →

D₁₇ #146

order	34
isoclinism	isoclinic class of D₁₇
automorphism	Aut(G) = AGL(1,𝔽₁₇) (order 272) · Inn(G) = G · Out(G) = C₈
holomorph	G ⋊ Aut(G) = #72213403
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇
unambiguous semidirect product	C₁₇⋊C₂

Splittings

Fitting, socle, derived, Sylow

C₁₇ ⋊ C₂

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 17

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17);(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)

Polycyclic presentation:

2,17|b^a=b^16

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	2	2	2	2	2	2	2	2	17
order	1	17	17	17	17	17	17	17	17	2
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	1	-1
χ₃	2	ζ+ζ¹⁶	ζ²+ζ¹⁵	ζ³+ζ¹⁴	ζ⁴+ζ¹³	ζ⁵+ζ¹²	ζ⁶+ζ¹¹	ζ⁷+ζ¹⁰	ζ⁸+ζ⁹	0
χ₄	2	ζ²+ζ¹⁵	ζ⁴+ζ¹³	ζ⁶+ζ¹¹	ζ⁸+ζ⁹	ζ⁷+ζ¹⁰	ζ⁵+ζ¹²	ζ³+ζ¹⁴	ζ+ζ¹⁶	0
χ₅	2	ζ³+ζ¹⁴	ζ⁶+ζ¹¹	ζ⁸+ζ⁹	ζ⁵+ζ¹²	ζ²+ζ¹⁵	ζ+ζ¹⁶	ζ⁴+ζ¹³	ζ⁷+ζ¹⁰	0
χ₆	2	ζ⁴+ζ¹³	ζ⁸+ζ⁹	ζ⁵+ζ¹²	ζ+ζ¹⁶	ζ³+ζ¹⁴	ζ⁷+ζ¹⁰	ζ⁶+ζ¹¹	ζ²+ζ¹⁵	0
χ₇	2	ζ⁵+ζ¹²	ζ⁷+ζ¹⁰	ζ²+ζ¹⁵	ζ³+ζ¹⁴	ζ⁸+ζ⁹	ζ⁴+ζ¹³	ζ+ζ¹⁶	ζ⁶+ζ¹¹	0
χ₈	2	ζ⁶+ζ¹¹	ζ⁵+ζ¹²	ζ+ζ¹⁶	ζ⁷+ζ¹⁰	ζ⁴+ζ¹³	ζ²+ζ¹⁵	ζ⁸+ζ⁹	ζ³+ζ¹⁴	0
χ₉	2	ζ⁷+ζ¹⁰	ζ³+ζ¹⁴	ζ⁴+ζ¹³	ζ⁶+ζ¹¹	ζ+ζ¹⁶	ζ⁸+ζ⁹	ζ²+ζ¹⁵	ζ⁵+ζ¹²	0
χ₁₀	2	ζ⁸+ζ⁹	ζ+ζ¹⁶	ζ⁷+ζ¹⁰	ζ²+ζ¹⁵	ζ⁶+ζ¹¹	ζ³+ζ¹⁴	ζ⁵+ζ¹²	ζ⁴+ζ¹³	0

Here ζ is a primitive root of unity of order 17 (ζ¹⁷ = 1).

Relationships

maximal subgroup of	#595, C₂×D₁₇, C₃×D₁₇, …
maximal normal subgroup of	#595, C₂×D₁₇, C₃×D₁₇, …
maximal quotient of	Dic₁₇, C₂×D₁₇, C₃×D₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34, 1);