C₁₇×#8865 #16866609

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

Aliases

direct product

C₁₇×#8865

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	#385
prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

348 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 2176

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

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

Conjugacy classes & character table

782 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 1853);