C₃×C₇×D₇ #64429

order	294
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂²×C₃×AGL(1,𝔽₇) (order 504) · Inn(G) = D₇ · Out(G) = C₂²×C₃²
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₇×D₇

	subgroup	G/H
center, hypercenter	C₃×C₇	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇²	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₇

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(294, 18);