C₃×#7398 #10626005

order	768
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7398

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#1343
commutator	C₂²×C₈	C₂×C₃×C₄
Frattini, cosocle	#408	C₂²×C₃
prime radical	C₃×#2018	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#408 ◁ G

258 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 42 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)(6,11,17,28)(8,13,22,30)(9,14,21,29)(10,23,24,15)(16,27,19,32)(18,25,20,33)(26,34,35,31);(5,10)(6,15)(7,19)(8,16)(11,13)(12,28)(14,26)(18,31)(21,22)(23,34)(24,32)(25,27)(29,33)(30,35)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 386);