C₁₇×#7381 #16865125

order	4352
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#7381

Subgroups

	subgroup	G/H
center	C₁₆×C₁₇	D₈
commutator	C₈	C₂×C₁₆×C₁₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₇
socle	C₂×C₁₇	D₄⋊C₁₆
prime radical	C₁₇×#135	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₇ ◁ C₂×C₁₆×C₁₇ ◁ C₄×C₁₆×C₁₇ ◁ G

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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)(18,19,22,28,24,30,47,59,23,31,46,60,21,40,43,71,25,32,48,61,44,27,54,56,42,29,62,58,45,57,73,79)(20,33,35,64,37,53,68,77,38,55,69,78,41,63,72,80,39,65,70,81,34,26,49,50,36,52,67,76,66,51,74,75);(18,20)(19,26)(21,41)(22,35)(23,38)(24,37)(25,39)(27,55)(28,50)(29,63)(30,52)(31,51)(32,53)(33,40)(34,44)(36,42)(43,72)(45,66)(46,69)(47,68)(48,70)(49,54)(56,78)(57,65)(58,80)(59,76)(60,75)(61,77)(62,67)(64,71)(73,74)(79,81)

Conjugacy classes & character table

1496 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 369);