C₃×#11577 #10630184

order	768
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#11577

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#99
commutator	C₂⁴	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×D₄ ◁ G

424 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,5,8,12,9,13,7,11)(6,10,14,17,15,18,16,19)(20,21)(22,26)(23,30,31,25)(24,27)(28,34,35,32)(29,33);(4,6)(5,10)(7,16)(8,14)(9,15)(11,19)(12,17)(13,18)(21,25,27,30)(23,31)(26,32,33,34)(28,35);(4,7,9,8)(5,11,13,12)(6,14,15,16)(10,17,18,19)(20,22,24,29)(21,26,27,33)(23,28,31,35)(25,32,30,34)

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4565);