C₃×C₇×#441 #13056952

order	1344
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×C₇×#441

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₇	C₂×D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ C₃×C₄²×C₇ ◁ G

216 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 30
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

462 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6241);