C₅×C₇₃ #67251

order	365
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄×C₈×C₉ (order 288) · Inn(G) = C₁ · Out(G) = C₄×C₈×C₉
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₆₅
abelian	C₅×C₇₃

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) = 78
transitive degree	μ_t(G) = 365

A minimal faithful permutation representation — 1 generator on 78 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)

Conjugacy classes & character table

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

Relationships

center of	C₅×D₃×C₇₃, C₅×D₅×C₇₃, C₅×D₁₉×C₇₃, …
derived subgroup of	D₃₆₅, Dic₃₆₅, #13091732, …
Fitting subgroup of	D₅×C₇₃, C₅×D₇₃, D₃₆₅, …
socle of	D₅×C₇₃, C₅×D₇₃, D₃₆₅, …
maximal subgroup of	D₅×C₇₃, C₅×D₇₃, D₃₆₅, …
maximal normal subgroup of	D₅×C₇₃, C₅×D₇₃, D₃₆₅, …
maximal quotient of	C₂×C₅×C₇₃, C₃×C₅×C₇₃, C₂₅×C₇₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(365, 1);

C5×C73 #67251