C₁₇×#12973 #16870717

order	4352
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#12973

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₇	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₇
prime radical	C₂³×C₁₇	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂³×C₁₇ ◁ C₂²×Q₈×C₁₇ ◁ G

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

normal subgroups only

This group has 460 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) = 1088

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)(24,30)(25,28)(26,35)(27,32)(29,33)(31,36)(34,37);(18,20,21,19)(23,28)(24,25)(26,36)(29,31)(30,32)(35,37);(22,24,27,25)(23,29,32,37)(26,34,36,33)(28,35,30,31)

Conjugacy classes & character table

680 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 5961);