C₄×C₈₃ #66521

order	332
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄₁ (order 164) · Inn(G) = C₁ · Out(G) = C₂²×C₄₁
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₃₂
abelian	C₄×C₈₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₈₃
socle, prime radical	C₂×C₈₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 332

A minimal faithful permutation representation — 1 generator on 87 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87)

Conjugacy classes & character table

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(332, 2);

center of	M₁₆×C₈₃, #41×C₈₃, C₃⋊C₈×C₈₃, …
derived subgroup of	#13050575, D₆₆₄, #13050577, …
Fitting subgroup of	C₈₃⋊C₈, Dic₁₆₆, C₄×D₈₃, …
maximal subgroup of	C₈₃⋊C₈, C₈×C₈₃, Dic₁₆₆, …
maximal normal subgroup of	C₈₃⋊C₈, C₈×C₈₃, Dic₁₆₆, …
maximal quotient of	C₈×C₈₃, C₂×C₄×C₈₃, Dic₃×C₈₃, …

C4×C83 #66521