C₃×#7272 #10625879

order	768
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7272

Subgroups

	subgroup	G/H
center	C₃×C₄	#333
commutator	C₄²	C₃×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃
socle	C₂×C₃	#1296
prime radical	C₂³×C₃	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

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

normal subgroups only

This group has 208 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) = 96

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

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

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, 260);