Finite Group Databasebeta

← Prev (#660) Next (#662) →

C₄×C₁₉ #661

order	76
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₉ (order 36) · Inn(G) = C₁ · Out(G) = C₂²×C₉
holomorph	G ⋊ Aut(G) = D₄×AGL(1,𝔽₁₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₇₆
abelian	C₄×C₁₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₁₉
socle, prime radical	C₂×C₁₉	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 1 generator on 23 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

center of	M₁₆×C₁₉, #41×C₁₉, C₁₉×C₃⋊C₈, …
derived subgroup of	#64516, D₁₅₂, Dic₇₆, …
Frattini subgroup of	D₁₄₄₄₀, Dic₇₂₂₀, Dic₁₀₁₀₈, …
Fitting subgroup of	C₁₉⋊C₈, Dic₃₈, C₄×D₁₉, …
maximal subgroup of	C₁₉⋊C₈, C₈×C₁₉, Dic₃₈, …
maximal normal subgroup of	C₁₉⋊C₈, C₈×C₁₉, Dic₃₈, …
maximal quotient of	C₈×C₁₉, C₂×C₄×C₁₉, Dic₃×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(76, 2);