C₁₇×#7407 #16865151

order	4352
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#7407

Subgroups

	subgroup	G/H
center	C₄×C₁₇	#353
commutator	C₂×M₁₆	C₂×C₄×C₁₇
Frattini, cosocle	#408	C₂²×C₁₇
socle	C₂×C₁₇	#1343
prime radical	C₁₇×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₁₇×#408 ◁ G

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

normal subgroups only

This group has 240 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)(18,19,22,27,24,28,41,56)(20,30,32,59,34,60,64,79)(21,26,37,50,38,52,70,54)(23,43,42,73,36,29,57,55)(25,45,46,67,47,74,75,35)(31,48,61,53,62,58,63,78)(33,66,65,81,44,49,40,72)(39,71,69,80,68,51,77,76);(18,20)(19,25)(21,35)(22,40)(23,44)(24,34)(26,49)(27,53)(28,47)(29,58)(30,51)(31,39)(32,57)(33,36)(37,63)(38,67)(41,65)(42,64)(43,48)(45,69)(46,55)(50,59)(52,66)(54,79)(56,78)(60,71)(61,70)(62,68)(72,80)(73,75)(74,77)(76,81)

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, 395);