C₃×#23565 #10642172

order	768
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#23565

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁵×C₃ ◁ 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) = 96

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

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

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, 16553);