D₇₆ #3888

order	152
isoclinism	isoclinic class of Dic₃₈
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₉) (order 2736) · Inn(G) = C₂×D₁₉ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₇₆
unambiguous semidirect product	C₄⋊D₁₉

Fitting	C₄×C₁₉ ⋊ C₂
Sylow	C₁₉ ⋊ D₄

16 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:

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

Polycyclic presentation:

2^3,19|b^2=c,b^a=bc,d^a=d^18

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(152, 5);