#6878⋊C₂ #92190

order	486
isoclinism	isoclinic class of #6878⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃

Aliases

Fitting, Sylow	#6878 ⋊ C₂
derived	C₉² ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉²	C₂×C₃
Frattini	C₃×C₉	C₃×D₃
Fitting	#6878	C₂
socle	C₃	#4179
LCS	G ▷ C₉²
UCS	C₁

35 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

2,3^5|c^3=e^2,d^3=f^2,c^a=c^2e,d^a=d^2f,e^a=e^2,f^a=f^2,c^b=cd,d^b=de,e^b=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 35);