(C₂×D₃)⋊Dic₇ #66574

order	336
isoclinism	isoclinic class of #4245
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

(C₂×D₃)⋊Dic₇

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×D₇
commutator	C₂×C₃×C₇	C₂×C₄
Fitting, socle, cosocle	C₂²×C₃×C₇	C₂²
prime radical	C₂²×D₃×C₇	C₂
hypocenter	C₃×C₇	C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂²

68 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

2^4,3,7|a^2=c,b^a=bd,f^a=f^6,e^b=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 43);