order	104
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₃×C₄×S₄ (order 288) · Inn(G) = C₂² · Out(G) = C₃×C₄×D₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₁₃)×#5935
homology	Abelianization (H₁) = C₂²×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

Q₈×C₁₃

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂×C₁₃	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₁₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ G

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21);(1,5,3,7)(2,6,4,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(104, 11);