C₁₇×#12277 #16870021

order	4352
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#12277

Subgroups

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

318 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 65
transitive degree	μ_t(G) = 1088

A minimal faithful permutation representation — 3 generators on 65 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,23,26)(20,27,29,32)(21,25,22,30)(24,31,33,28)(34,35)(36,40)(37,47)(38,51)(39,43)(41,50)(42,52)(44,59)(45,58)(46,55)(48,63)(49,62)(53,60)(54,61)(56,64)(57,65);(18,20)(19,24)(21,28)(22,31)(23,29)(25,32)(26,33)(27,30)(34,36,39,46)(35,40,43,55)(37,48,49,44)(38,45,52,61)(41,56,57,53)(42,58,51,54)(47,59,62,63)(50,64,65,60);(18,21,23,22)(19,25,26,30)(20,28,29,31)(24,32,33,27)(34,37)(35,41,43,57)(36,44)(38,50)(39,49)(40,53,55,56)(42,47,51,62)(45,60)(46,48)(52,65)(54,59,58,63)(61,64)

Conjugacy classes & character table

425 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 5265);