order	158
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₁₃ (order 78) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₁₃
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₇₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₅₈
abelian	C₂×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) = 81
transitive degree	μ_t(G) = 158

A minimal faithful permutation representation — 1 generator on 81 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)

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₇₉, Q₈×C₇₉, Dic₃×C₇₉, …
derived subgroup of	Dic₁₅₈, D₃₁₆, C₂²⋊D₇₉, …
Frattini subgroup of	C₄×C₆₂₄₁, Dic₆₂₄₁, Dic₁₂₄₈₂, …
Fitting subgroup of	Dic₇₉, C₂×D₇₉, C₂×C₇₉⋊C₃, …
socle of	Dic₇₉, C₄×C₇₉, C₂×D₇₉, …
maximal subgroup of	Dic₇₉, C₄×C₇₉, C₂×D₇₉, …
maximal normal subgroup of	Dic₇₉, C₄×C₇₉, C₂×D₇₉, …
maximal quotient of	C₄×C₇₉, C₂²×C₇₉, D₃×C₇₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(158, 2);

C2×C79 #3924

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×C₇₉ #3924