C₃×#7700 #10626307

order	768
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#7700

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×C₄³ ◁ G

394 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 768

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 688);