C₃×#19084 #10637691

order	768
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#19084

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃
prime radical	C₂³×C₃	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

824 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 384

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 12072);