C₃×#7257 #10625864

order	768
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7257

Subgroups

	subgroup	G/H
center	C₃×C₄	#327
commutator	C₂×C₈	C₃×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂²×C₃
socle	C₂×C₃	#1294
prime radical	C₂⁴×C₃	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×M₁₆ ◁ G

302 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 98 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)(4,5,8,13,10,14,9,25)(6,15,17,30,19,34,18,33)(7,20,22,32,24,31,23,12)(11,26,27,35,29,21,28,16);(4,6,8,17,10,19,9,18)(5,11,13,27,14,29,25,28)(7,21,23,35,24,26,22,16)(12,30,31,15,32,33,20,34)

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, 245);