H(3,𝔽₃)×C₅₉ #13097571

order	1593
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₅₉ · Schur multiplier (H₂) = C₃²

Aliases

direct product	H(3,𝔽₃)×C₅₉
unambiguous semidirect product	C₃²⋊(C₃×C₅₉)
unambiguous semidirect product	(C₃²×C₅₉)⋊C₃

Subgroups

	subgroup	G/H
center, socle	C₃×C₅₉	C₃²
commutator, Frattini, cosocle	C₃	C₃²×C₅₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅₉ ◁ G

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 531

A minimal faithful permutation representation — 2 generators on 68 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅₉×AΣL(1,𝔽₂₇), C₅₉×#731, C₅₉×#732, …
maximal normal subgroup of	C₅₉×AΣL(1,𝔽₂₇), C₅₉×#731, C₅₉×#732, …
maximal quotient of	C₅₉×C₃²⋊C₉, C₅₉×AΣL(1,𝔽₂₇), C₅₉×#731, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1593, 3);

H(3,𝔽3)×C59 #13097571