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C₈×C₁₃ #1056

order	104
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₄ (order 48) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₄
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/8)×AGL(1,𝔽₁₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₀₄
abelian	C₈×C₁₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₁₃
socle, prime radical	C₂×C₁₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

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

104 conjugacy classes — view the full character table.

Relationships

center of	C₁₃×M₃₂, C₁₃×#131, C₁₃×C₃⋊C₁₆, …
derived subgroup of	D₂₀₈, #88040, Dic₁₀₄, …
Frattini subgroup of	D₁₃₅₂₀, Dic₆₇₆₀, Dic₉₄₆₄, …
Fitting subgroup of	#6090, C₈×D₁₃, #6094, …
maximal subgroup of	#6090, C₁₃×C₁₆, C₈×D₁₃, …
maximal normal subgroup of	#6090, C₁₃×C₁₆, C₈×D₁₃, …
maximal quotient of	C₁₃×C₁₆, C₂×C₈×C₁₃, C₁₃×C₃⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(104, 2);