C₇×C₂²⋊(C₄²⋊C₃) #13060797

order	1344
isoclinism	isoclinic class of C₂²⋊(C₄²⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product	C₇×C₂²⋊(C₄²⋊C₃)
unambiguous semidirect product	C₂²⋊(C₇×C₄²⋊C₃)

Fitting, Hall	C₂²×C₄²×C₇ ⋊ C₃
derived, Sylow	C₂²×C₄² ⋊ C₃×C₇

	subgroup	G/H
center, hypercenter	C₇	C₂²⋊(C₄²⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₄²	C₃×C₇
Frattini	C₂²	C₇×C₂²⋊A₄
Fitting	C₂²×C₄²×C₇	C₃
socle	C₂⁴×C₇	A₄
LCS	G ▷ C₂²×C₄²
UCS	C₁ ◁ C₇

206 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 168

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10086);