C₁₆×Q₁₆×C₁₇ #16869796

order	4352
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₆×Q₁₆×C₁₇

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₁₇	D₄
commutator	C₄	C₂²×C₁₆×C₁₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₇
socle, prime radical	C₂²×C₁₇	C₈×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₇ ◁ C₄×C₁₆×C₁₇ ◁ 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) = 49
transitive degree	μ_t(G) = 4352

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(17,18,19,20,21,22,23,24)(25,32,31,30,29,28,27,26);(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32)

Conjugacy classes & character table

1904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 5040);