C₂²⋊D₇ #288

order	56
isoclinism	isoclinic class of Dic₁₄
automorphism	Aut(G) = C₂²×AGL(1,𝔽₇) (order 168) · Inn(G) = C₂×D₇ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72214429
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

C₂²⋊D₇

Fitting	C₂²×C₇ ⋊ C₂
Sylow	C₇ ⋊ D₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2,4,6,7,5,3)(9,11);(2,3)(4,5)(6,7)(8,9)(10,11)

Polycyclic presentation:

2^3,7|b^a=bc,d^a=d^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(56, 7);