H(3,𝔽₃)×C₄₇ #11929644

order	1269
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄₇ · Schur multiplier (H₂) = C₃²

Aliases

direct product	H(3,𝔽₃)×C₄₇
unambiguous semidirect product	C₃²⋊(C₃×C₄₇)
unambiguous semidirect product	(C₃²×C₄₇)⋊C₃

Subgroups

	subgroup	G/H
center, socle	C₃×C₄₇	C₃²
commutator, Frattini, cosocle	C₃	C₃²×C₄₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄₇ ◁ G

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 423

A minimal faithful permutation representation — 2 generators on 56 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);(2,5,6)(4,8,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₇×AΣL(1,𝔽₂₇), C₄₇×#731, C₄₇×#732, …
maximal normal subgroup of	C₄₇×AΣL(1,𝔽₂₇), C₄₇×#731, C₄₇×#732, …
maximal quotient of	C₄₇×C₃²⋊C₉, C₄₇×AΣL(1,𝔽₂₇), C₄₇×#731, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1269, 3);

H(3,𝔽3)×C47 #11929644