order	237
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₁₃ (order 156) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₁₃
holomorph	G ⋊ Aut(G) = D₃×AGL(1,𝔽₇₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₂₃₇
abelian	C₃×C₇₉

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) = 82
transitive degree	μ_t(G) = 237

A minimal faithful permutation representation — 1 generator on 82 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82)

Conjugacy classes & character table

237 conjugacy classes — view the full character table.

Relationships

center of	C₃×D₃×C₇₉, H(3,𝔽₃)×C₇₉, 3₋¹⁺²×C₇₉, …
derived subgroup of	D₂₃₇, Dic₂₃₇, D₃×D₇₉, …
Fitting subgroup of	D₃×C₇₉, C₃×D₇₉, D₂₃₇, …
socle of	D₃×C₇₉, C₃×D₇₉, D₂₃₇, …
maximal subgroup of	D₃×C₇₉, C₃×D₇₉, D₂₃₇, …
maximal normal subgroup of	D₃×C₇₉, C₃×D₇₉, D₂₃₇, …
maximal quotient of	C₂×C₃×C₇₉, C₉×C₇₉, C₃²×C₇₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(237, 2);

C3×C79 #6634

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×C₇₉ #6634