C₃×#9402 #10628009

order	768
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃×#9402

Subgroups

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

486 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 138 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)(12,13,18,21,17,22,16,20)(14,23,25,19,26,27,24,15);(4,5,9,11,7,10,6,8)(12,14)(13,19)(15,22)(16,25)(17,26)(18,24)(20,27)(21,23);(5,10)(8,11)(12,15,17,19)(13,14,22,26)(16,23,18,27)(20,25,21,24)

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, 2390);