C₄×D₈×C₅₉ #16836971

order	3776
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₄×D₈×C₅₉

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₉	D₄
commutator	C₄	C₂²×C₄×C₅₉
Frattini, cosocle	C₂×C₄	C₂³×C₅₉
socle	C₂²×C₅₉	C₂×D₄
prime radical	C₂×D₈×C₅₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₄²×C₅₉ ◁ G

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 1888

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3,4)(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);(5,6,7,8,9,10,11,12);(6,12)(7,11)(8,10)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅₉×#1331, C₅₉×#1333, C₈×D₈×C₅₉, …
maximal normal subgroup of	C₅₉×#1331, C₅₉×#1333, C₈×D₈×C₅₉, …
maximal quotient of	C₈×D₈×C₅₉, C₅₉×#1581, C₅₉×#1589, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3776, 118);

C4×D8×C59 #16836971