C₄×C₇₁ #63343

order	284
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₅×C₇ (order 140) · Inn(G) = C₁ · Out(G) = C₂²×C₅×C₇
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₂₈₄
abelian	C₄×C₇₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₇₁
socle, prime radical	C₂×C₇₁	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 284

A minimal faithful permutation representation — 1 generator on 75 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)

Conjugacy classes & character table

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(284, 2);

center of	M₁₆×C₇₁, #41×C₇₁, C₃⋊C₈×C₇₁, …
derived subgroup of	#11765850, D₅₆₈, #11765852, …
Fitting subgroup of	C₇₁⋊C₈, Dic₁₄₂, C₄×D₇₁, …
maximal subgroup of	C₇₁⋊C₈, C₈×C₇₁, Dic₁₄₂, …
maximal normal subgroup of	C₇₁⋊C₈, C₈×C₇₁, Dic₁₄₂, …
maximal quotient of	C₈×C₇₁, C₂×C₄×C₇₁, Dic₃×C₇₁, …

C4×C71 #63343