C₄×C₇₉ #64654

order	316
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₁₃ (order 156) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₁₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₁₆
abelian	C₄×C₇₉
multiplicative	(ℤ/317)ˣ, (ℤ/634)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₇₉
socle, prime radical	C₂×C₇₉	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 316

A minimal faithful permutation representation — 1 generator on 83 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83)

Conjugacy classes & character table

316 conjugacy classes. The character table is too large to display.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(316, 2);

center of	M₁₆×C₇₉, #41×C₇₉, C₃⋊C₈×C₇₉, …
derived subgroup of	#11929609, D₆₃₂, #11929611, …
Fitting subgroup of	C₇₉⋊C₈, Dic₁₅₈, C₄×D₇₉, …
maximal subgroup of	C₇₉⋊C₈, C₈×C₇₉, Dic₁₅₈, …
maximal normal subgroup of	C₇₉⋊C₈, C₈×C₇₉, Dic₁₅₈, …
maximal quotient of	C₈×C₇₉, C₂×C₄×C₇₉, Dic₃×C₇₉, …
automorphism group of	C₃₁₇, C₂×C₃₁₇
outer automorphism group of	C₃₁₇, C₂×C₃₁₇

C4×C79 #64654