C₃×#21795 #10640402

order	768
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₃×#21795

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₃×C₄
Frattini, cosocle	C₂⁴	C₂⁴×C₃
prime radical	C₂×C₃×#534	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₂²⋊C₄ ◁ G

1538 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 366 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) = 96

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 14783);