C₃×#8533 #10627140

order	768
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#8533

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#379
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

656 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 192

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1521);