C₃×#9500 #10628107

order	768
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃×#9500

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#386
commutator	C₂⁴	C₂²×C₃×C₄
Frattini, cosocle	C₂⁵	C₂³×C₃
prime radical	C₂⁵×C₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁵×C₃ ◁ G

660 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 96

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

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

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2488);