C₁₇×#2222 #14525228

order	2176
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#2222

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₇
prime radical	#41×C₁₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₁₇×#131 ◁ G

140 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 60 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)(18,19,25,31)(20,32,38,60)(21,39,42,56)(22,43,45,54)(23,30,24,29)(26,47,51,35)(27,46,53,41)(28,44,55,33)(34,63,66,80)(36,59,37,58)(40,71,72,52)(48,75,76,81)(49,67,50,61)(57,68,79,65)(62,70,73,78)(64,69,74,77);(18,20,33,61,24,37,45,47,25,38,44,67,23,36,22,35)(19,26,43,59,30,50,55,60,31,51,54,58,29,49,28,32)(21,34,62,81,41,65,72,74,42,66,73,75,46,68,40,64)(27,48,70,63,39,69,71,79,53,76,78,80,56,77,52,57);(18,21)(19,27)(20,34)(22,40)(23,46)(24,41)(25,42)(26,48)(28,52)(29,56)(30,39)(31,53)(32,57)(33,62)(35,64)(36,68)(37,65)(38,66)(43,70)(44,73)(45,72)(47,74)(49,77)(50,69)(51,76)(54,78)(55,71)(58,80)(59,63)(60,79)(61,81)(67,75)

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16955020, #16955027, #16955192, …
maximal subgroup of	C₁₇×#13671, C₁₇×#13672, C₁₇×#13701, …
maximal normal subgroup of	C₁₇×#13671, C₁₇×#13672, C₁₇×#13701, …
maximal quotient of	C₁₇×#12431, C₁₇×#12434, C₁₇×#12440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 954);