C₃×#9398 #10628005

order	768
nilpotency class	3
isoclinism	isoclinic class of #9383
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#9398

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#386
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂×C₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

574 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(5,8)(6,9)(10,11)(12,13,16,20,17,19,26,14)(15,22,24,27,23,18,25,21);(4,5,9,11,7,10,6,8)(12,14,16,19,17,20,26,13)(15,21,25,22,23,27,24,18);(12,15,17,23)(13,18,19,22)(14,21,20,27)(16,25,26,24)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2386);