C₃×#12243 #10630850

order	768
nilpotency class	4
isoclinism	isoclinic class of #12148
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#12243

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₃
prime radical	C₃×#392	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂×C₃×C₂²⋊C₄ ◁ G

464 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 74 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)(4,5,7,10)(6,8,11,9)(12,13,17,21)(14,22,24,27)(15,20,16,19)(18,25,26,23);(5,8)(6,11)(9,10)(12,14)(13,18)(15,23)(16,25)(17,24)(19,22)(20,27)(21,26);(12,15)(13,19,21,20)(14,23)(16,17)(18,22,26,27)(24,25)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5231);