C₃⋊(C₃⋊D₂₁) #67406

order	378
isoclinism	isoclinic class of C₃⋊(C₃⋊D₂₁)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃³

Aliases

unambiguous semidirect product	C₃⋊(C₃⋊D₂₁)
unambiguous semidirect product	C₇⋊(C₃⋊(C₃⋊D₃))

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³×C₇	C₂
LCS	G ▷ C₃³×C₇
UCS	C₁

112 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 189

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 59);