C₁₇×#7439 #16865183

order	4352
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#7439

Subgroups

	subgroup	G/H
center	C₄×C₁₇	#353
commutator	C₂×M₁₆	C₂×C₄×C₁₇
Frattini, cosocle	#408	C₂²×C₁₇
socle	C₂×C₁₇	#1352
prime radical	C₂×#41×C₁₇	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₁₇×#408 ◁ G

170 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 81
transitive degree	μ_t(G) = 1088

A minimal faithful permutation representation — 2 generators on 81 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,27,24,28,40,55)(20,30,32,56,33,57,62,79)(21,26,36,49,37,51,66,53)(23,29,41,54,42,48,70,76)(25,43,44,35,46,63,72,64)(31,45,59,52,60,73,61,78)(34,47,39,68,58,74,69,81)(38,50,65,77,67,75,71,80);(18,20)(19,25)(21,35,37,64)(22,39,40,69)(23,34)(24,33)(26,47,51,74)(27,52,55,78)(28,46)(29,45)(30,50,57,75)(31,38,60,67)(32,41,62,70)(36,59)(42,58)(43,71)(44,54,72,76)(48,73)(49,56)(53,79)(61,66)(63,65)(68,77)(80,81)

Conjugacy classes & character table

527 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 427);