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C₃×C₁₉ #296

order	57
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) = D₃×AGL(1,𝔽₁₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₅₇
abelian	C₃×C₁₉

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) = 22
transitive degree	μ_t(G) = 57

A minimal faithful permutation representation — 1 generator on 22 points:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

center of	C₃×D₃×C₁₉, C₁₉×H(3,𝔽₃), C₁₉×3₋¹⁺², …
derived subgroup of	D₅₇, Dic₅₇, D₃×D₁₉, …
Frattini subgroup of	C₉×C₃₆₁, #15692423, C₅×C₉×C₃₆₁, …
Fitting subgroup of	D₃×C₁₉, C₃×D₁₉, D₅₇, …
socle of	D₃×C₁₉, C₃×D₁₉, D₅₇, …
maximal subgroup of	D₃×C₁₉, C₃×D₁₉, D₅₇, …
maximal normal subgroup of	D₃×C₁₉, C₃×D₁₉, D₅₇, …
maximal quotient of	C₂×C₃×C₁₉, C₉×C₁₉, C₃²×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(57, 2);