C₇×#61317 #13238533

order	1792
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₇ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

2064 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 5 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 54305);