C₃×#1638 #72353

order	384
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#1638

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃
prime radical	C₂×C₃×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

194 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#1638, #10770958, #10771211, …
maximal subgroup of	C₂×C₃×#1638, C₃×#18494, C₃×#18521, …
maximal normal subgroup of	C₂×C₃×#1638, C₃×#18494, C₃×#18521, …
maximal quotient of	C₃×#7765, C₃×#7767, C₃×#7773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4929);