C₃×#12709 #10631316

order	768
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#12709

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#1781
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×Q₈	C₂³×C₃
prime radical	C₃×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ C₂²×C₃×#41 ◁ G

986 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 35 points:

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

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, 5697);