C₃×#13646 #10632253

order	768
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#13646

Subgroups

	subgroup	G/H
center	C₂×C₃×C₁₆	C₂³
commutator	C₂²	C₂²×C₃×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₃
socle	C₂²×C₃	C₂²×C₁₆
prime radical	C₂×C₃×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₁₆ ◁ G

202 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3)(4,5)(6,7)(8,9,12,17,13,18,26,33,14,10,19,21,27,22,34,36,11,15,23,28,24,29,37,38,25,20,16,30,31,35,32,39);(5,7)(9,15)(10,20)(17,28)(18,29)(21,30)(22,35)(33,38)(36,39);(8,10,12,21,13,22,26,36,14,15,19,28,27,29,34,38,11,20,23,30,24,35,37,39,25,9,16,17,31,18,32,33)

Conjugacy classes & character table

336 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6634);