(C₂²⋊D₁₉)⋊C₃ #90691

order	456
isoclinism	isoclinic class of (C₁₉⋊C₃)⋊Q₈
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₉) (order 1368) · Inn(G) = C₂×D₁₉⋊C₃ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

(C₂²⋊D₁₉)⋊C₃

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,8)(3,12,16)(4,6,13)(5,15,7)(9,18,14)(10,17,19)(21,23);(2,3)(4,6)(5,9)(7,8)(10,14)(11,16)(12,18)(13,19)(15,17)(20,21)(22,23)

Polycyclic presentation:

2^2,3,2,19|b^a=bd,e^a=e^18,e^c=e^7

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 11);