C₄×Q₁₆×C₅₉ #16836973

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

Aliases

direct product

C₄×Q₁₆×C₅₉

Subgroups

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

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 3776

A minimal faithful permutation representation — 3 generators on 79 points:

(1,2,3,4)(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);(5,6,7,8,9,10,11,12)(13,20,19,18,17,16,15,14);(5,13,9,17)(6,14,10,18)(7,15,11,19)(8,16,12,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅₉×#1332, C₅₉×#1334, C₈×Q₁₆×C₅₉, …
maximal normal subgroup of	C₅₉×#1332, C₅₉×#1334, C₈×Q₁₆×C₅₉, …
maximal quotient of	C₈×Q₁₆×C₅₉, C₅₉×#1584, C₅₉×#1591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3776, 120);

C4×Q16×C59 #16836973