C₃×C₈×C₅₃ #11929655

order	1272
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³×C₄×C₁₃ (order 416) · Inn(G) = C₁ · Out(G) = C₂³×C₄×C₁₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₇₂
abelian	C₃×C₈×C₅₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃×C₅₃
socle, prime radical	C₂×C₃×C₅₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1272

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

1272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₁₆×C₅₃, C₂×C₃×C₈×C₅₃, C₈×C₉×C₅₃, …
maximal normal subgroup of	C₃×C₁₆×C₅₃, C₂×C₃×C₈×C₅₃, C₈×C₉×C₅₃, …
maximal quotient of	C₃×C₁₆×C₅₃, C₂×C₃×C₈×C₅₃, C₈×C₉×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1272, 4);

C3×C8×C53 #11929655