C₇×#627 #92684

order	504
isoclinism	isoclinic class of #625
automorphism	Aut(G) = C₂×C₃×#64241 (order 1728) · Inn(G) = D₃² · Out(G) = C₂×C₃×D₄
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

Hall	C₃²×C₇ ⋊ Q₈
Sylow	C₃² ⋊ C₇×Q₈

54 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 110);