C₄⋊#157 #3790

order	144
isoclinism	isoclinic class of #3788
automorphism	Aut(G) = D₄×AΓL(1,𝔽₉) (order 1152) · Inn(G) = C₂×#157 · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

Fitting	C₃²×C₄ ⋊ C₄
Sylow	C₃² ⋊ C₄⋊C₄

42 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 10 points:

(1,3,6)(2,5,4)(7,8,9,10);(1,2)(3,5,6,4)(8,10)

Polycyclic presentation:

2^4,3^2|a^2=c,b^2=d,b^a=bd,e^a=ef^2,f^a=e^2f^2,e^c=e^2,f^c=f^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 133);