C₃×#23925 #10642532

order	768
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₃×#23925

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×#41 ◁ G

2080 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 2080 conjugacy classes of subgroups — too many to draw. Only its 302 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)(6,7)(12,13)(14,17)(15,18)(16,20)(19,23)(21,25)(22,24)(26,27);(5,7)(8,9)(10,11)(12,14,16,21)(13,17,20,25)(15,22,23,27)(18,24,19,26);(9,11)(13,18)(15,23)(17,24)(19,20)(22,27)(25,26);(5,7)(14,21)(17,25)(22,27)(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, 16913);