C₃×#1664 #72379

order	384
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#1664

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

184 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#1664, #10776363, #10776390, …
maximal subgroup of	C₂×C₃×#1664, C₃×#18987, C₃×#18990, …
maximal normal subgroup of	C₂×C₃×#1664, C₃×#18987, C₃×#18990, …
maximal quotient of	C₃×#7764, C₃×#7765, C₃×#7766, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4955);