C₃×#20979 #10639586

order	768
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₃×#20979

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₂²⋊C₄
prime radical	C₂×C₃×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂⁴×C₃×C₄ ◁ G

1954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(4,5)(6,7)(8,9,12,15,13,16,11,21)(10,17,20,22,18,23,19,14);(8,10)(9,14,16,22)(11,19)(12,20)(13,18)(15,17,21,23);(5,7);(10,18)(14,22)(17,23)(19,20)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#20979, #72249256, #72253682, …
maximal subgroup of	D₃×#20979, C₃×#359639, #72249256, …
maximal normal subgroup of	D₃×#20979, C₃×#359639, #72249256, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13967);

C3×#20979 #10639586

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#20979 #10639586