C₃×#12697 #10631304

order	768
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₃×#12697

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#99
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×D₄	C₂³×C₃
prime radical	C₂×C₃×#535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

1182 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(4,5)(6,7)(8,9)(10,14)(11,15,19,13)(12,16)(17,22)(18,21,23,20);(9,13)(10,17)(11,19)(14,21)(15,16)(20,22);(5,7)(8,10,12,17)(9,14,16,22)(11,18,19,23)(13,20,15,21)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#12697, #72270478
maximal subgroup of	C₃×#493372, D₃×#12697, #72270478
maximal normal subgroup of	C₃×#493372, D₃×#12697, #72270478

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5685);