C₇⋊#157 #6993

order	252
isoclinism	isoclinic class of C₇⋊#157
automorphism	Aut(G) = AGL(1,𝔽₇)×AΓL(1,𝔽₉) (order 6048) · Inn(G) = G · Out(G) = C₃×D₄
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₇	C₄
prime radical, cosocle	C₇×C₃⋊D₃	C₂
LCS	G ▷ C₃²×C₇
UCS	C₁

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

2^2,3^2,7|a^2=b,c^a=c^2d,d^a=cd,e^a=e^6,c^b=c^2,d^b=d^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 32);