C₂×C₃×C₅₃ #64661

order	318
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄×C₁₃ (order 104) · Inn(G) = C₁ · Out(G) = C₂×C₄×C₁₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₁₈
abelian	C₂×C₃×C₅₃

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) = 58
transitive degree	μ_t(G) = 318

A minimal faithful permutation representation — 1 generator on 58 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)

Conjugacy classes & character table

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(318, 4);

center of	C₃×D₄×C₅₃, C₃×Q₈×C₅₃, C₃×Dic₃×C₅₃, …
derived subgroup of	#11929661, #11929662, #11929663, …
Fitting subgroup of	Dic₃×C₅₃, C₃×Dic₅₃, Dic₁₅₉, …
socle of	Dic₃×C₅₃, C₃×Dic₅₃, Dic₁₅₉, …
maximal subgroup of	Dic₃×C₅₃, C₃×Dic₅₃, Dic₁₅₉, …
maximal normal subgroup of	Dic₃×C₅₃, C₃×Dic₅₃, Dic₁₅₉, …
maximal quotient of	C₃×C₄×C₅₃, C₂²×C₃×C₅₃, C₂×C₉×C₅₃, …

C2×C3×C53 #64661