C₃×#7350 #10625957

order	768
nilpotency class	5
isoclinism	isoclinic class of #7348
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7350

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#1317
commutator	C₂²×C₄	C₂×C₃×C₈
Frattini, cosocle	#407	C₂²×C₃
prime radical	C₂⁴×C₃	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#407 ◁ G

178 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 35 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 338);