(C₂²⋊D₇)⋊C₄ #6444

order	224
automorphism	Aut(G) = C₂⁶×AGL(1,𝔽₇) (order 2688) · Inn(G) = C₂²×D₇ · Out(G) = C₂⁴×C₃
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

(C₂²⋊D₇)⋊C₄

Fitting	C₇×C₂²⋊C₄ ⋊ C₂
Sylow	C₇ ⋊ C₄×D₄

94 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:

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

Polycyclic presentation:

2^5,7|c^2=d,b^a=be,f^a=f^6,c^b=ce

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 76);