C₃×#7068 #10625675

order	768
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7068

Subgroups

	subgroup	G/H
center	C₂×C₃×C₈	D₈
commutator	C₈	C₂×C₃×C₁₆
Frattini, cosocle	C₈²	C₂²×C₃
socle	C₂²×C₃	D₄⋊C₈
prime radical	C₂×C₃×D₈	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈ ◁ C₃×C₈² ◁ G

162 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 70 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)(6,9)(7,16)(8,12)(10,13)(11,14)(15,18)(17,19)(20,21,22,25,23,26,28,31,24,27,29,32,30,33,34,35);(5,9)(6,13)(7,14)(10,11)(12,18)(15,19)(16,17)

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 56);