(C₃⋊D₃)⋊D₇ #6998

order	252
isoclinism	isoclinic class of (C₃⋊D₃)⋊D₇
automorphism	Aut(G) = AGL(1,𝔽₇)×GO⁺(4,𝔽₂) (order 3024) · Inn(G) = G · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	(C₃⋊D₃)⋊D₇
unambiguous semidirect product	D₂₁⋊D₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₇	C₂²
LCS	G ▷ C₃²×C₇
UCS	C₁

44 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)(5,6)(7,8,10,12,13,11,9);(2,3)(4,5,6)(8,9)(10,11)(12,13)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 37);