C₃²×Q₁₆ #3765

order	144
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = AGL(1,ℤ/8)×GL(2,𝔽₃) (order 1536) · Inn(G) = D₄ · Out(G) = C₂²×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₃

Aliases

C₃²×Q₁₆

	subgroup	G/H
center, socle, prime radical	C₂×C₃²	D₄
commutator, Frattini, cosocle	C₄	C₂²×C₃²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄ ◁ G

54 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 108);