C₁₇×#12764 #16870508

order	4352
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₇×#12764

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₇	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×D₄	C₂³×C₁₇
prime radical	C₂²×D₄×C₁₇	C₂³
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂³×C₁₇ ◁ C₂²×D₄×C₁₇ ◁ G

768 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 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)(24,32,35,47)(25,29)(26,39,41,30)(27,31)(28,34)(33,50,36,49)(37,48,53,45)(38,46)(40,43)(42,51)(44,52);(22,24,27,35)(23,28,31,43)(25,36,38,33)(26,40,41,34)(29,44,46,42)(30,47,39,32)(37,51,53,52)(45,50,48,49);(19,21)(22,25,27,38)(23,29,31,46)(24,33,35,36)(26,37,41,53)(28,42,43,44)(30,48,39,45)(32,49,47,50)(34,51,40,52)

Conjugacy classes & character table

578 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 5752);