C₃×#60400 #10679007

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

Aliases

direct product

C₃×#60400

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

4732 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 4732 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)(10,21)(22,32)(23,33)(34,35);(4,5)(6,10)(7,11)(8,12)(9,13)(14,22)(15,23)(16,21)(17,24)(18,25)(19,26)(20,27)(28,34)(29,32)(30,35)(31,33);(4,6)(5,10)(7,14)(8,15)(9,16)(11,22)(12,23)(13,21)(17,28)(18,29)(19,30)(20,31)(24,34)(25,32)(26,35)(27,33);(4,7,9,18)(5,11,13,25)(6,14,16,29)(8,19,20,17)(10,22,21,32)(12,26,27,24)(15,30,31,28)(23,35,33,34);(4,8,9,20)(5,12,13,27)(6,15,16,31)(7,17,18,19)(10,23,21,33)(11,24,25,26)(14,28,29,30)(22,34,32,35)

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, 53388);