C₃×#29710 #10648317

order	768
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃×#29710

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
prime radical	C₃×#3459	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

1116 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 242 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)(5,6)(9,10)(12,13)(14,17)(15,18)(16,19)(20,25)(21,24)(22,26)(23,27);(4,5)(6,8)(7,9)(10,11)(12,14)(13,17)(15,20)(16,21)(18,25)(19,24)(22,27)(23,26);(4,6,8,5)(7,9,11,10)(12,15,16,22)(13,18,19,26)(14,20,21,27)(17,23,24,25);(4,7,8,11)(5,9,6,10)

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, 22698);