C₃×#8746 #10627353

order	768
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#8746

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#381
commutator	C₂⁴	C₂²×C₃×C₄
Frattini, cosocle	C₂⁵	C₂³×C₃
prime radical	C₂³×C₃×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁵×C₃ ◁ G

814 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 27 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1734);