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C₄×C₁₇ #594

order	68
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₁₆ (order 32) · Inn(G) = C₁ · Out(G) = C₂×C₁₆
holomorph	G ⋊ Aut(G) = D₄×AGL(1,𝔽₁₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₆₈
abelian	C₄×C₁₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₁₇
socle, prime radical	C₂×C₁₇	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 68

A minimal faithful permutation representation — 1 generator on 21 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

center of	M₁₆×C₁₇, #41×C₁₇, C₁₇×C₃⋊C₈, …
derived subgroup of	#63219, D₁₃₆, Dic₆₈, …
Frattini subgroup of	Dic₈₀₉₂, D₁₆₁₈₄, Dic₁₂₇₁₆, …
Fitting subgroup of	#3622, Dic₃₄, C₄×D₁₇, …
maximal subgroup of	#3622, C₈×C₁₇, Dic₃₄, …
maximal normal subgroup of	#3622, C₈×C₁₇, Dic₃₄, …
maximal quotient of	C₈×C₁₇, C₂×C₄×C₁₇, Dic₃×C₁₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(68, 2);