order	216
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×D₄×C₉ (order 144) · Inn(G) = C₂² · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₂²×C₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₄×C₂₇
unambiguous semidirect product	C₄⋊(C₂×C₂₇)

Subgroups

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 2 generators on 31 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31);(2,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 10);