C₃×#90017 #13055678

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#90017

Splittings

Fitting	C₃×C₇×2₋¹⁺⁴ ⋊ C₂
Hall	C₃×C₇ ⋊ #454
Sylow	C₇ ⋊ C₃×#454

Subgroups

	subgroup	G/H
center	C₂×C₃	#6516
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×2₋¹⁺⁴	C₂
socle	C₂×C₃×C₇	#120
hypercenter	C₃×2₋¹⁺⁴	D₇
hypocenter	C₇	C₃×#454
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×2₋¹⁺⁴

292 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 3 generators on 26 points:

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

Conjugacy classes & character table

201 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4967);