C₃×#60402 #10679009

order	768
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₃×#60402

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×Q₈ ◁ G

4778 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 1114 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 — 5 generators on 35 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53390);