C₄₉ #251

order	49
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₇ (order 42) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₇
holomorph	G ⋊ Aut(G) = #72215772
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₄₉

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₇	C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₇ ▷ C₁

3 conjugacy classes of subgroups.

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 49

A minimal faithful permutation representation — 1 generator on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(49, 1);