C₃×#8188 #10626795

order	768
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#8188

Subgroups

	subgroup	G/H
center	C₃×C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	C₄×C₂²⋊C₄
prime radical	C₂×C₃×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

412 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 182 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 — 3 generators on 35 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1176);