C₃×C₇×#560 #13061980

order	1344
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×C₇×#560

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ G

504 conjugacy classes of subgroups; only the 284 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 284 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) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

399 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11269);