C₇×C₄²⋊C₃ #66587

order	336
isoclinism	isoclinic class of C₄²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product	C₇×C₄²⋊C₃
unambiguous semidirect product	C₄²⋊(C₃×C₇)

Fitting, Hall	C₄²×C₇ ⋊ C₃
derived, Sylow	C₄² ⋊ C₃×C₇

	subgroup	G/H
center, hypercenter	C₇	C₄²⋊C₃
commutator, cosocle, hypocenter	C₄²	C₃×C₇
Frattini	C₂²	C₇×A₄
Fitting	C₄²×C₇	C₃
socle	C₂²×C₇	A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₇

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 56);