C₂×C₃×C₅₉ #67058

order	354
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₂₉ (order 116) · Inn(G) = C₁ · Out(G) = C₂²×C₂₉
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₅₄
abelian	C₂×C₃×C₅₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 354

A minimal faithful permutation representation — 1 generator on 64 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)

Conjugacy classes & character table

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(354, 4);

center of	C₃×D₄×C₅₉, C₃×Q₈×C₅₉, C₃×Dic₃×C₅₉, …
derived subgroup of	#13083462, #13083463, #13083464, …
Fitting subgroup of	Dic₃×C₅₉, C₃×Dic₅₉, Dic₁₇₇, …
socle of	Dic₃×C₅₉, C₃×Dic₅₉, Dic₁₇₇, …
maximal subgroup of	Dic₃×C₅₉, C₃×Dic₅₉, Dic₁₇₇, …
maximal normal subgroup of	Dic₃×C₅₉, C₃×Dic₅₉, Dic₁₇₇, …
maximal quotient of	C₃×C₄×C₅₉, C₂²×C₃×C₅₉, C₂×C₉×C₅₉, …

C2×C3×C59 #67058