C₂×D₇² #87661

order	392
isoclinism	isoclinic class of D₇²
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×D₇²
unambiguous semidirect product	(C₇×D₇)⋊C₂²
unambiguous semidirect product	(C₇⋊D₇)⋊C₂²

Fitting, socle	C₂×C₇² ⋊ C₂²
derived, Sylow	C₇² ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂	D₇²
commutator, cosocle, hypocenter	C₇²	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₇²	C₂²
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂

79 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 3 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(392, 41);