D₇⋊C₂₇ #67348

order	378
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	Aut(G) = C₉×AGL(1,𝔽₇) (order 378) · Inn(G) = AGL(1,𝔽₇) · Out(G) = C₉
homology	Abelianization (H₁) = C₂×C₂₇ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	D₇⋊C₂₇
unambiguous semidirect product	(C₇⋊C₂₇)⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₉	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂×C₂₇
Fitting, cosocle	C₇×C₉	C₂×C₃
socle	C₃×C₇	C₂×C₉
prime radical	C₃×D₇	C₉
LCS	G ▷ C₇
UCS	C₁ ◁ C₉

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 189

A minimal faithful permutation representation — 2 generators on 34 points:

(2,6,5,3,4,7)(8,13,25,16,28,27,30,9,18,11,21,31,24,33,12,15,14,26,17,29,34,10,20,19,23,22,32);(1,2,5,6,4,7,3)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 1);