C₃×#19056 #10637663

order	768
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃×#19056

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃
prime radical	C₃×#3549	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 992 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) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3)(4,5)(6,14)(7,11)(8,12)(9,13)(10,16)(15,18)(17,19)(20,21,25,28)(22,29,31,34)(23,27,24,32)(26,33,35,30);(4,6)(5,10)(7,15)(8,16)(9,17)(11,14)(12,19)(13,18)(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32);(4,6)(5,10)(7,16)(8,15)(9,17)(11,19)(12,14)(13,18)(23,24)(27,32)(29,34)(30,33);(4,7,9,8)(5,11,13,12)(6,15,17,16)(10,14,18,19)(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29)

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, 12044);