C₇×#543⋊C₃ #13060798

order	1344
automorphism	?
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₇×#543⋊C₃
unambiguous semidirect product	#543⋊(C₃×C₇)

Fitting, Hall	C₇×#543 ⋊ C₃
derived, Sylow	#543 ⋊ C₃×C₇

	subgroup	G/H
center, socle, hypercenter	C₂²×C₇	C₂²⋊A₄
commutator, cosocle, hypocenter	#543	C₃×C₇
Frattini	C₂²	C₇×C₂²⋊A₄
Fitting	C₇×#543	C₃
LCS	G ▷ #543
UCS	C₁ ◁ C₂²×C₇

144 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5,6,7)(9,10,14)(13,16,20)(15,18,17)(21,23,22);(8,14,12,17)(9,18,15,10)(11,20,19,22)(13,23,21,16);(9,13)(10,16)(15,21)(18,23)

maximal subgroup of	ΓL(1,𝔽₈)×#543⋊C₃, C₇×#11710730, D₇×#543⋊C₃
maximal normal subgroup of	ΓL(1,𝔽₈)×#543⋊C₃, D₇×#543⋊C₃

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10087);