C₂×#346 #2106

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×#72238006 (order 4096) · Inn(G) = C₂² · Out(G) = C₂×#449475
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

100 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 838);