C₇×#5245 #13056280

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#5245

Splittings

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

Subgroups

	subgroup	G/H
center	C₂×C₇	#953
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×2₋¹⁺⁴	C₂
socle	C₂×C₃×C₇	#120
hypercenter	C₇×2₋¹⁺⁴	D₃
hypocenter	C₃	C₇×#454
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×2₋¹⁺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:

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

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5569);