C₂×C₃³×Q₁₆ #11724015

order	864
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂²×C₃³

Aliases

C₂×C₃³×Q₁₆

	subgroup	G/H
center, socle, prime radical	C₂²×C₃³	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ C₂×C₃³×C₄ ◁ G

840 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 864

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2)(19,20,21);(3,4,5,6,7,8,9,10)(11,18,17,16,15,14,13,12)(22,23,24);(3,11,7,15)(4,12,8,16)(5,13,9,17)(6,14,10,18)(25,26,27)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 3982);