C₃⋊D₃×#10624749 #72244336

order	13122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₃⋊D₃×#10624749

Splittings

Fitting, Sylow

C₃²×#10624749 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃⋊D₃×C₃²⋊H(3,𝔽₃)
commutator, cosocle	C₃⁵	C₂×C₃³
Frattini	C₃³	C₃³×C₃⋊D₃
Fitting	C₃²×#10624749	C₂
socle	C₃³	C₂×C₃²⋊H(3,𝔽₃)
hypercenter	#10624749	C₃⋊D₃
hypocenter	C₃²	C₂×#10624749
LCS	G ▷ C₃⁵ ▷ C₃³ ▷ C₃²
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #10624749

23568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 33 points:

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

Conjugacy classes & character table

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

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C3⋊D3×#10624749 #72244336