C₄²×D₇ #6434

order	224
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = AGL(1,𝔽₇)×#87524 (order 16128) · Inn(G) = D₇ · Out(G) = C₃×#87524
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₄²×D₇

Fitting	C₄²×C₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₄²

108 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 66);