C₁₇×#7382 #16865126

order	4352
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#7382

Subgroups

	subgroup	G/H
center	C₈×C₁₇	#102
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 ◁ C₈×C₁₇ ◁ C₂×C₁₆×C₁₇ ◁ C₄×C₁₆×C₁₇ ◁ G

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

normal subgroups only

This group has 118 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)(18,19,22,29,24,28,45,56,25,30,46,59,21,38,40,68,26,31,47,60,41,57,71,80,23,48,44,75,42,61,72,81)(20,32,34,62,36,27,50,51,35,53,67,77,39,54,69,78,37,64,43,73,33,55,65,58,49,63,74,79,66,52,70,76);(18,20)(19,27)(21,39)(22,43)(23,49)(24,36)(25,35)(26,37)(28,53)(29,58)(30,54)(31,55)(32,61)(33,41)(34,47)(38,64)(40,70)(42,66)(44,67)(45,65)(46,74)(48,52)(50,71)(51,60)(56,79)(57,63)(59,76)(62,68)(69,72)(73,81)(75,78)(77,80)

Conjugacy classes & character table

1088 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 370);