C₃×#13648 #10632255

order	768
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#13648

Subgroups

	subgroup	G/H
center	C₃×C₁₆	C₂×D₄
commutator	C₄	C₂²×C₃×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₃
socle	C₂×C₃	D₄×C₁₆
prime radical	C₃×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₁₆ ◁ C₂×C₃×C₃₂ ◁ G

184 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3)(4,5,9,16,10,17,34,49,11,15,35,47,37,48,40,65,12,18,36,50,38,20,52,54,8,30,32,61,33,53,58,64)(6,19,22,51,23,28,56,60,21,13,39,41,55,29,14,44,24,42,46,66,57,26,45,59,31,43,62,67,63,7,25,27);(4,6)(5,13)(7,20)(8,31)(9,22)(10,23)(11,21)(12,24)(14,40)(15,42)(16,41)(17,29)(18,43)(19,30)(25,58)(26,48)(27,54)(28,53)(32,62)(33,63)(34,56)(35,39)(36,46)(37,55)(38,57)(44,49)(45,52)(47,66)(50,67)(51,61)(59,65)(60,64);(4,7,9,27,10,19,34,51,11,28,35,60,37,13,40,41,12,29,36,44,38,42,52,66,8,26,32,59,33,43,58,67)(5,14,16,24,17,46,49,57,15,45,47,31,48,62,65,63,18,25,50,6,20,22,54,23,30,56,61,21,53,39,64,55)

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#13648, #72254973
maximal subgroup of	C₃×#151103, D₃×#13648, #72254973
maximal normal subgroup of	C₃×#151103, D₃×#13648, #72254973

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6636);