C₁₆×Q₁₆×C₆₁ #29515485

order	15616
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₆×Q₁₆×C₆₁

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₆₁	D₄
commutator	C₄	C₂²×C₁₆×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
socle, prime radical	C₂²×C₆₁	C₈×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₆₁ ◁ C₄×C₁₆×C₆₁ ◁ G

218 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 15616

A minimal faithful permutation representation — 3 generators on 93 points:

(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)

Conjugacy classes & character table

6832 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5040);