C₃×#12028 #10630635

order	768
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#12028

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#106
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₂³×C₃	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

210 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 35 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5016);