C₃×#11556 #10630163

order	768
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#11556

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#99
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₃
prime radical	C₂⁴×C₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#117 ◁ G

366 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 98 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)(4,5,7,10,8,11,14,17)(6,9,13,16,12,15,18,19)(20,21,24,29,25,30,22,27)(23,28,31,34,33,35,32,26);(4,6)(5,9)(7,13)(8,12)(10,16)(11,15)(14,18)(17,19)(20,22,25,24)(21,26,30,34)(23,32,33,31)(27,28,29,35);(6,12)(9,15)(13,18)(16,19)(20,22,25,24)(21,27,30,29)(23,31,33,32)(26,28,34,35)

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4544);