Dic₉⋊C₄ #3669

order	144
isoclinism	isoclinic class of Dic₁₈
automorphism	Aut(G) = C₂⁴×AGL(1,ℤ/9) (order 864) · Inn(G) = C₂×D₉ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

Dic₉⋊C₄

C₉ ⋊ C₄⋊C₄

	subgroup	G/H
center	C₂²	C₂×D₉
commutator	C₂×C₉	C₂×C₄
Frattini, socle, prime radical	C₂²×C₃	C₂×D₃
Fitting	C₂×C₄×C₉	C₂
cosocle	C₂²×C₉	C₂²
hypercenter	C₂×C₄	D₉
hypocenter	C₉	C₄⋊C₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄

39 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 12);