C₃×#11676 #10630283

order	768
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#11676

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×#102
prime radical	C₂³×C₃	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

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

normal subgroups only

This group has 316 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) = 43
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3)(4,5,7,10,8,6,11,9)(12,13)(14,23)(15,20)(16,29)(17,33)(18,22)(19,25)(21,31)(24,37)(26,43)(27,40)(28,38)(30,36)(32,39)(34,41)(35,42);(4,6)(5,8)(7,9)(10,11)(12,14,18,27)(13,19,22,35)(15,26,17,24)(16,30,32,41)(20,36,33,34)(21,37,38,43)(23,39,40,29)(25,28,42,31);(5,6)(9,10)(12,15,18,17)(13,20,22,33)(14,24,27,26)(16,28,32,31)(19,34,35,36)(21,29,38,39)(23,37,40,43)(25,41,42,30)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4664);