C₁₇×#12112 #16869856

order	4352
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#12112

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
prime radical	C₂×#41×C₁₇	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₁₇×#408 ◁ G

390 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 89
transitive degree	μ_t(G) = 2176

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(22,25)(26,27,33,39)(28,40,44,68)(29,45,48,65)(30,49,53,63)(31,38,32,37)(34,56,59,42)(35,54,61,47)(36,51,64,52)(41,69,73,86)(43,74,55,62)(46,77,80,60)(50,66,81,58)(57,78,85,89)(67,72,88,71)(70,75,83,84)(76,82,87,79);(18,19)(20,22)(21,23)(24,25)(26,28,33,44)(27,34,39,59)(29,41,48,73)(30,50,53,81)(31,43,32,55)(35,57,61,85)(36,62,64,74)(37,58,38,66)(40,63,68,49)(42,51,56,52)(45,75,65,84)(46,78,80,89)(47,71,54,72)(60,86,77,69)(67,87,88,76)(70,82,83,79);(18,20)(19,22)(21,24)(23,25)(26,29)(27,35)(28,41)(30,46)(31,54)(32,47)(33,48)(34,57)(36,60)(37,65)(38,45)(39,61)(40,67)(42,70)(43,72)(44,73)(49,76)(50,78)(51,82)(52,79)(53,80)(55,71)(56,83)(58,84)(59,85)(62,86)(63,87)(64,77)(66,75)(68,88)(69,74)(81,89)

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, 5100);