C₃×#11575 #10630182

order	768
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#11575

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#99
commutator	C₂⁴	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×D₄ ◁ G

554 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 27 points:

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

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, 4563);