C₇⋊C₂₇ #4424

order	189
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₉×AGL(1,𝔽₇) (order 378) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×C₉
homology	Abelianization (H₁) = C₂₇ · Schur multiplier (H₂) = C₁

Aliases

C₇⋊C₂₇

C₇ ⋊ C₂₇

	subgroup	G/H
center, Frattini, hypercenter	C₉	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₂₇
Fitting, cosocle	C₇×C₉	C₃
socle, prime radical	C₃×C₇	C₉
LCS	G ▷ C₇
UCS	C₁ ◁ C₉

8 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(189, 1);