C₃×#29659 #10648266

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

Aliases

direct product

C₃×#29659

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

974 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 4 generators on 35 points:

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

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, 22647);