C₃×#19067 #10637674

order	768
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#19067

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃
prime radical	C₂³×C₃	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

784 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 51
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 4 generators on 51 points:

(1,2,3)(4,5,9,6)(7,14,15,19)(8,13,18,11)(10,17,12,16)(20,21)(22,32)(23,37)(24,29)(25,30)(26,31)(27,35)(28,39)(33,41)(34,42)(36,45)(38,43)(40,44)(46,48)(47,50)(49,51);(4,6,9,5)(7,12,15,10)(8,11,18,13)(14,17,19,16)(20,22)(21,27)(23,33)(24,34)(25,35)(26,36)(28,41)(29,32)(30,45)(31,42)(37,50)(38,47)(39,48)(40,49)(43,51)(44,46);(4,7)(5,10)(6,12)(8,17)(9,15)(11,19)(13,14)(16,18)(20,23,26,40)(21,28,31,43)(22,33,36,49)(24,39,25,38)(27,41,42,51)(29,44,30,37)(32,46,45,50)(34,48,35,47);(20,24,26,25)(21,29,31,30)(22,34,36,35)(23,38,40,39)(27,32,42,45)(28,37,43,44)(33,47,49,48)(41,50,51,46)

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, 12055);