C₁₇×#7468 #16865212

order	4352
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#7468

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₇	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆×C₁₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₁₇
socle	C₂²×C₁₇	C₂²⋊C₁₆
prime radical	C₂³×C₁₇	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₇ ◁ C₂²×C₁₆×C₁₇ ◁ G

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

normal subgroups only

This group has 182 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) = 2176

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)(22,23,24,29,25,30,33,41,26,31,34,42,35,43,46,51,27,28,32,38,36,39,44,48,37,40,45,49,47,50,52,53);(18,19,20,21)(23,28)(24,32)(30,39)(31,40)(33,44)(34,45)(43,50)(46,52)

Conjugacy classes & character table

1496 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 456);