C₃×#29657 #10648264

order	768
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#29657

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁵×C₃ ◁ G

1148 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

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

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, 22645);