D₄×D₇×S₄ #72244666

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

D₄×D₇×S₄

Fitting	C₂²×C₇×D₄ ⋊ C₂×D₃
Sylow	C₇ ⋊ C₂×D₄×S₄

2590 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 4 generators on 15 points:

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

maximal subgroup of	D₇×S₄², D₄×S₄×AGL(1,𝔽₇)
maximal normal subgroup of	D₄×S₄×AGL(1,𝔽₇)

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,4)(5,7,8)(9,10,12,14,15,13,11), (1,2)(3,4)(6,7), (10,11)(12,13)(14,15), (6,7));