C₂×C₄×Dic₇ #6485

order	224
isoclinism	isoclinic class of D₇
automorphism	Aut(G) order 21504
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product	C₂×C₄×Dic₇
unambiguous semidirect product	(C₂×C₇)⋊C₄²

C₇ ⋊ C₂×C₄²

108 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 117);