D₄×Q₈×3₋¹⁺² #13154631

order	1728
nilpotency class	2
isoclinism	isoclinic class of D₄²×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵

Aliases

D₄×Q₈×3₋¹⁺²

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃²
prime radical	C₂×C₃²×D₄	C₂²×C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

1120 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20694);