C₃×#7095 #10625702

order	768
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×#7095

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	#331
commutator	C₄×C₈	C₂×C₃×C₄
Frattini, cosocle	C₈²	C₂²×C₃
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 768

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 83);