C₃×#3531 #87415

order	384
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₃×#3531

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
socle	C₂²×C₃	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

930 conjugacy classes of subgroups; only the 760 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 760 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 — 5 generators on 35 points:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10687625, #10776385, #10776397, …
maximal subgroup of	C₃×#8505, C₃×#18990, C₃×#18991, …
maximal normal subgroup of	C₃×#8505, C₃×#18990, C₃×#18991, …
maximal quotient of	C₃×#34348, C₃×#34350, C₃×C₄×#562, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19991);