H(3,𝔽₃)×C₄₁ #11764303

order	1107
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄₁ · Schur multiplier (H₂) = C₃²

Aliases

direct product	H(3,𝔽₃)×C₄₁
unambiguous semidirect product	C₃²⋊(C₃×C₄₁)
unambiguous semidirect product	(C₃²×C₄₁)⋊C₃

Subgroups

	subgroup	G/H
center, socle	C₃×C₄₁	C₃²
commutator, Frattini, cosocle	C₃	C₃²×C₄₁
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄₁ ◁ G

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 369

A minimal faithful permutation representation — 2 generators on 50 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);(2,5,6)(4,8,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₁×AΣL(1,𝔽₂₇), C₄₁×#731, C₄₁×#732, …
maximal normal subgroup of	C₄₁×AΣL(1,𝔽₂₇), C₄₁×#731, C₄₁×#732, …
maximal quotient of	C₄₁×C₃²⋊C₉, C₄₁×AΣL(1,𝔽₂₇), C₄₁×#731, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1107, 3);

H(3,𝔽3)×C41 #11764303