C₅₉×#443 #16836977

order	3776
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₉×#443

Subgroups

	subgroup	G/H
center	C₈×C₅₉	D₄
commutator	C₄	C₂²×C₄×C₅₉
Frattini, cosocle	C₂×C₄	C₂³×C₅₉
socle	C₂×C₅₉	C₄×D₄
prime radical	#135×C₅₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₉ ◁ C₂×C₈×C₅₉ ◁ G

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 944

A minimal faithful permutation representation — 3 generators 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)(60,61,64,68,65,62,69,67)(63,66,71,74,72,70,75,73);(60,62)(61,65)(63,70)(64,67)(66,72)(68,69)(71,73)(74,75);(60,63,64,71,65,72,69,75)(61,66,68,74,62,70,67,73)

Conjugacy classes & character table

1652 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₉×#1335, C₅₉×#1336, C₅₉×#2170, …
maximal normal subgroup of	C₅₉×#1335, C₅₉×#1336, C₅₉×#2170, …
maximal quotient of	C₈×D₈×C₅₉, C₈×SD₁₆×C₅₉, C₈×Q₁₆×C₅₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3776, 124);

C59×#443 #16836977