C₃×#7044 #10625651

order	768
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7044

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	#95
commutator	C₂²×C₄	C₃×C₄²
Frattini, cosocle	C₂²×C₄²	C₂²×C₃
prime radical	C₂⁴×C₃	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄² ◁ G

352 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 384

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 32);