C₁₇×#7401 #16865145

order	4352
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#7401

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1343
commutator	C₂²×C₈	C₂×C₄×C₁₇
Frattini, cosocle	#408	C₂²×C₁₇
prime radical	C₂×#41×C₁₇	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₁₇×#408 ◁ G

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

normal subgroups only

This group has 234 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)(18,19,22,27,24,28,42,57)(20,30,32,60,34,61,65,75)(21,35,38,55,40,26,49,52)(23,29,43,56,37,51,70,74)(25,45,46,36,47,68,76,69)(31,48,63,54,64,58,77,80)(33,50,66,78,44,67,41,72)(39,59,71,81,62,53,73,79);(18,20,24,34)(19,25,28,47)(21,36,40,69)(22,41,42,66)(23,44,37,33)(26,50,35,67)(27,54,57,80)(29,58,51,48)(30,59,61,53)(31,62,64,39)(32,43,65,70)(38,63,49,77)(45,73,68,71)(46,74,76,56)(52,75,55,60)(72,79,78,81)

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, 389);