C₃×#11675 #10630282

order	768
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#11675

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×#102
prime radical	C₃×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

360 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 384

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4663);