C₃×#7226 #10625833

order	768
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7226

Subgroups

	subgroup	G/H
center	C₃×C₄	H(3,ℤ/4)
commutator	C₂×C₈	C₃×C₄²
Frattini, cosocle	C₄×M₁₆	C₂²×C₃
socle	C₂×C₃	#1288
prime radical	C₂³×C₃	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ G

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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)(6,14,16,27)(7,18,19,32)(8,12)(9,13)(10,15)(11,20,25,33)(17,28,31,35)(21,24)(22,29)(23,30)(26,34);(4,6)(5,10,12,22,13,23,24,26)(7,15)(8,17)(9,16)(11,14,18,28,25,27,32,35)(19,30)(20,29)(21,31)(33,34)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 214);