C₅₉×D₄⋊C₈ #16836859

order	3776
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₅₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₅₉×D₄⋊C₈
unambiguous semidirect product	D₄⋊(C₈×C₅₉)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₉	D₄
commutator	C₄	C₂×C₈×C₅₉
Frattini, cosocle	C₄²	C₂²×C₅₉
socle	C₂²×C₅₉	C₂²⋊C₄
prime radical	C₂×D₄×C₅₉	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₄²×C₅₉ ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 1888

A minimal faithful permutation representation — 2 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)(62,64)(63,66)(65,67)(68,69,70,72,71,73,74,75);(61,64)(62,67)(65,66)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₅₉×D₄⋊C₈, C₅₉×#1476, C₅₉×#1477, …
maximal normal subgroup of	C₂×C₅₉×D₄⋊C₈, C₅₉×#1476, C₅₉×#1477, …
maximal quotient of	C₅₉×#1270, C₅₉×#1271, C₅₉×#1279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3776, 6);

C59×D4⋊C8 #16836859