C₃×#7057 #10625664

order	768
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7057

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#99
commutator	C₂×C₄	C₂×C₃×C₁₆
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₃
socle	C₂²×C₃	#323
prime radical	C₂³×C₃	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₈ ◁ G

196 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 62 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)(6,12,14,19)(7,15,17,10)(8,11)(9,13)(16,18)(20,21,22,25,23,26,28,31,24,27,29,32,30,33,34,35);(4,6,8,14)(5,9,11,18)(7,16,17,13)(10,19,15,12)

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, 45);