C₇×#5065 #13056100

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#5065

Splittings

Hall	C₃×C₇ ⋊ #499
Sylow	C₃ ⋊ C₇×#499

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#125	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₇×#125	D₃
hypocenter	C₃	C₇×#499
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#125

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

normal subgroups only

This group has 172 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) = 34
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3,4,5,6,7)(8,9,10)(11,13,14,18)(12,15,16,17)(20,25)(21,28)(22,30)(24,31)(26,27)(32,33);(11,12,14,16)(13,17,18,15)(19,21)(20,24)(22,29)(23,28)(25,33)(26,31)(27,32)(30,34);(9,10)(19,20,22,26,23,27,30,25)(21,24,29,31,28,32,34,33)

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5389);