C₃×#20340 #10638947

order	768
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#20340

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂²×C₄²
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	C₂²×#95
prime radical	C₂²×C₃×#41	C₂²
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ G

1956 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 1114 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) = 96

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

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13328);