C₁₇×#12966 #16870710

order	4352
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₇×#12966

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	C₂×C₁₇×#127	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂³×C₁₇ ◁ C₂²×D₄×C₁₇ ◁ G

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

normal subgroups only

This group has 580 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) = 37
transitive degree	μ_t(G) = 544

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21,20)(22,23)(24,27)(25,28)(26,30)(29,34)(31,36)(32,37)(33,35);(18,20,21,19)(22,24)(23,27)(25,33)(26,31)(28,37)(29,35)(30,36)(32,34);(23,28)(24,31)(25,34)(27,35)(29,30)(36,37)

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, 5954);