C₉×C₇₃ #10620322

order	657
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₈×C₉ (order 432) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₈×C₉
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₆₅₇
abelian	C₉×C₇₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₃×C₇₃
socle, prime radical	C₃×C₇₃	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 657

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

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(657, 2);

center of	C₉×D₁₁×C₇₃, C₉×D₁₃×C₇₃, C₇₃×#6869, …
derived subgroup of	D₆₅₇, Dic₆₅₇, C₉⋊#64408, …
Fitting subgroup of	D₉×C₇₃, C₉×D₇₃, D₆₅₇, …
maximal subgroup of	D₉×C₇₃, C₉×D₇₃, D₆₅₇, …
maximal normal subgroup of	D₉×C₇₃, C₉×D₇₃, D₆₅₇, …
maximal quotient of	C₂×C₉×C₇₃, C₂₇×C₇₃, C₃×C₉×C₇₃, …

C9×C73 #10620322