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C₂×C₁₇ #147

order	34
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₁₆ (order 16) · Inn(G) = C₁ · Out(G) = C₁₆
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₁₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₄
abelian	C₂×C₁₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 34

A minimal faithful permutation representation — 1 generator on 19 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₁₇, Q₈×C₁₇, Dic₃×C₁₇, …
derived subgroup of	Dic₃₄, D₆₈, C₂²⋊D₁₇, …
Frattini subgroup of	Dic₂₈₉, C₄×C₂₈₉, C₃×C₄×C₂₈₉, …
Fitting subgroup of	Dic₁₇, C₂×D₁₇, #3624, …
socle of	Dic₁₇, C₄×C₁₇, C₂×D₁₇, …
maximal subgroup of	Dic₁₇, C₄×C₁₇, C₂×D₁₇, …
maximal normal subgroup of	Dic₁₇, C₄×C₁₇, C₂×D₁₇, …
maximal quotient of	C₄×C₁₇, C₂²×C₁₇, D₃×C₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34, 2);