C₇×#4914 #13055949

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4914

Splittings

Fitting	C₃×C₇×SD₃₂ ⋊ C₂
Hall	C₃×C₇ ⋊ #509
Sylow	C₃ ⋊ C₇×#509

Subgroups

	subgroup	G/H
center	C₂×C₇	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³×C₇
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×SD₃₂	C₂
socle	C₂×C₃×C₇	C₂×D₈
hypercenter	C₇×SD₃₂	D₃
hypocenter	C₃	C₇×#509
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×SD₃₂

180 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 62 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)(11,12,15,16)(13,20,22,26)(14,23,21,18)(17,25,24,19);(12,17)(13,19)(14,21)(16,24)(18,26)(20,23)(22,25);(9,10)(12,16)(17,24)(18,23)(20,26)

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5238);