C₁₇×#12684 #16870428

order	4352
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₇×#12684

Subgroups

	subgroup	G/H
center	C₈×C₁₇	#120
commutator	C₂³	C₂²×C₈×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
socle	C₂×C₁₇	#1779
prime radical	C₁₇×#585	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ G

894 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 210 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 — 3 generators on 81 points:

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

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, 5672);