C₃×#7113 #10625720

order	768
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7113

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	#102
commutator	C₂×C₈	C₂×C₃×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

254 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 768

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 101);