C₃×#21796 #10640403

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×#21796

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×#537	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄⋊C₄ ◁ G

1334 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1334 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) = 192

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

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

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, 14784);