C₅²×C₃⋊(C₃⋊D₃) #13062550

order	1350
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₅² · Schur multiplier (H₂) = C₃³×C₅

Aliases

direct product	C₅²×C₃⋊(C₃⋊D₃)
unambiguous semidirect product	C₃⋊(C₅²×C₃⋊D₃)

Fitting, socle, Hall	C₃³×C₅² ⋊ C₂
derived, Sylow	C₃³ ⋊ C₂×C₅²

	subgroup	G/H
center, hypercenter	C₅²	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂×C₅²
Frattini	C₁	G
Fitting, socle	C₃³×C₅²	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₅²

448 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 675

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2,3,4,5)(12,13)(15,16)(18,19);(6,7,8,9,10)(17,18,19);(14,15,16);(11,12,13)

375 conjugacy classes. The character table is too large to display.

maximal subgroup of	C₅⋊D₅×C₃⋊(C₃⋊D₃), C₅²×D₃×C₃⋊D₃, #72222444, …
maximal normal subgroup of	C₅⋊D₅×C₃⋊(C₃⋊D₃), C₅²×D₃×C₃⋊D₃, #72222444, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1350, 109);