C₇×C₂²⋊(C₂²⋊A₄) #13062426

order	1344
isoclinism	isoclinic class of C₂²⋊(C₂²⋊A₄)
automorphism	?
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product	C₇×C₂²⋊(C₂²⋊A₄)
unambiguous semidirect product	C₂²⋊(C₇×C₂²⋊A₄)

Fitting, socle, Hall	C₂⁶×C₇ ⋊ C₃
derived, Sylow	C₂⁶ ⋊ C₃×C₇

	subgroup	G/H
center, hypercenter	C₇	C₂²⋊(C₂²⋊A₄)
commutator, cosocle, hypocenter	C₂⁶	C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂⁶×C₇	C₃
LCS	G ▷ C₂⁶
UCS	C₁ ◁ C₇

2030 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11715);