(C₁₃⋊C₃)⋊Q₈ #64592

order	312
isoclinism	isoclinic class of (C₁₃⋊C₃)⋊Q₈
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₃) (order 1248) · Inn(G) = C₂×D₁₃⋊C₃ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	(C₁₃⋊C₃)⋊Q₈
unambiguous semidirect product	Dic₂₆⋊C₃

24 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:

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

Polycyclic presentation:

2^2,3,2,13|a^2=d,b^2=d,b^a=bd,e^a=e^12,e^c=e^3

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 8);