C₃×#7373 #10625980

order	768
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#7373

Subgroups

	subgroup	G/H
center	C₃×C₄	#353
commutator	C₄²	C₂×C₃×C₈
Frattini, cosocle	#423	C₂²×C₃
socle	C₂×C₃	#1325
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

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

normal subgroups only

This group has 152 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)(4,5,7,12,8,13,22,29,9,14,23,30,16,11,21,27)(6,15,17,28,18,10,20,24,19,25,33,34,31,26,32,35);(4,6,16,31,9,19,8,18)(5,10,13,25,14,26,11,15)(7,20,22,33,23,32,21,17)(12,28,27,35,30,34,29,24)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 361);