C₃×#12047 #10630654

order	768
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#12047

Subgroups

	subgroup	G/H
center	C₃×C₈	C₂×D₈
commutator	C₈	C₂²×C₃×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₃
socle	C₂×C₃	C₈×D₈
prime radical	C₃×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₁₆ ◁ C₃×C₄×C₁₆ ◁ G

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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)(4,5,9,15,11,16,28,14,12,17,8,27,10,6,18,20)(7,21,23,35,25,19,33,34,26,13,22,30,24,31,32,29);(4,6)(5,11)(7,19)(8,14)(9,20)(10,17)(12,16)(13,25)(15,28)(18,27)(21,24)(22,29)(23,34)(26,31)(30,33)(32,35);(4,7,9,23,11,25,28,33,12,26,8,22,10,24,18,32)(5,13,15,30,16,31,14,29,17,21,27,35,6,19,20,34)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72245453, #72245577, #72254957
maximal subgroup of	C₃×#483512, D₃×#12047, #72226252, …
maximal normal subgroup of	C₃×#483512, D₃×#12047, #72226252, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5035);