C₇×#5143 #13056178

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

Aliases

direct product

C₇×#5143

Splittings

Hall	C₃×C₇ ⋊ #473
Sylow	C₃ ⋊ C₇×#473

Subgroups

	subgroup	G/H
center	C₂×C₇	#930
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#131	C₂
socle	C₂×C₃×C₇	#121
prime radical	C₃×C₇×#41	C₂²
hypercenter	C₇×#131	D₃
hypocenter	C₃	C₇×#473
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#131

200 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 42
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3,4,5,6,7)(8,9,10)(11,14)(12,19)(13,24)(15,31)(16,29)(17,30)(18,32)(20,36)(21,28)(22,35)(23,37)(25,41)(26,39)(27,40)(33,42)(34,38);(11,13,15,25,17,27,16,26)(12,18,20,33,22,34,21,23)(14,24,31,41,30,40,29,39)(19,32,36,42,35,38,28,37);(9,10)(11,12,17,22)(13,23,27,33)(14,28,30,36)(15,21,16,20)(18,25,34,26)(19,31,35,29)(24,38,40,32)(37,41,42,39)

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, 5467);