C₃×#7369 #10625976

order	768
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7369

Subgroups

	subgroup	G/H
center	C₃×C₄	#354
commutator	C₄²	C₂×C₃×C₈
Frattini, cosocle	#423	C₂²×C₃
socle	C₂×C₃	#1324
prime radical	C₂³×C₃	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#423 ◁ G

132 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 46 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)(4,5,7,13,9,12,22,29,10,15,23,32,8,14,24,31)(6,16,18,33,20,11,25,26,17,27,34,30,19,28,21,35);(4,6,8,19,10,17,9,20)(5,11,14,16,15,28,12,27)(7,21,22,18,23,25,24,34)(13,30,29,35,32,33,31,26)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 357);