C₁₇×#7467 #16865211

order	4352
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#7467

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₁₇	D₄
commutator	C₄	C₄×C₁₆×C₁₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₁₇
socle	C₂²×C₁₇	C₄⋊C₁₆
prime radical	C₂³×C₁₇	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₇ ◁ C₂²×C₁₆×C₁₇ ◁ G

190 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 53
transitive degree	μ_t(G) = 1088

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23,25,31,26,33,36,46,27,30,37,43,39,44,49,52,28,32,38,45,40,47,50,53,24,29,34,41,35,42,48,51);(19,21)(23,29,32,30)(31,41,45,43)(33,42,47,44)(46,51,53,52)

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, 455);