C₈×C₅₃ #88323

order	424
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄×C₁₃ (order 208) · Inn(G) = C₁ · Out(G) = C₂²×C₄×C₁₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₂₄
abelian	C₈×C₅₃

Subgroups

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

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 424

A minimal faithful permutation representation — 1 generator on 61 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)

Conjugacy classes & character table

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

Relationships

center of	M₃₂×C₅₃, #131×C₅₃, C₅₃×#350, …
derived subgroup of	#13133158, D₈₄₈, #13133160, …
Fitting subgroup of	#11719908, C₈×D₅₃, #11719912, …
maximal subgroup of	#11719908, C₁₆×C₅₃, C₈×D₅₃, …
maximal normal subgroup of	#11719908, C₁₆×C₅₃, C₈×D₅₃, …
maximal quotient of	C₁₆×C₅₃, C₂×C₈×C₅₃, C₃⋊C₈×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(424, 2);

C8×C53 #88323