C₂×C₃×C₄⋊C₈ #5296

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×#347042 (order 2048) · Inn(G) = C₂² · Out(G) = C₂³×B(4,𝔽₂)
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₃×C₄⋊C₈

138 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 855);