C₇×#5141 #13056176

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

Aliases

direct product

C₇×#5141

Splittings

Fitting	C₃×C₇×#131 ⋊ C₂
Hall	C₃×C₇ ⋊ #471
Sylow	C₃ ⋊ C₇×#471

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	C₇×#949	C₂
hypercenter	C₇×#131	D₃
hypocenter	C₃	C₇×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#131

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

normal subgroups only

This group has 208 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) = 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)(12,17)(14,20)(15,21)(16,23)(18,26)(19,22)(24,25);(11,12,14,16,15,18,22,24)(13,17,20,23,21,26,19,25);(9,10)(12,16)(13,19)(14,22)(18,24)(20,21)(23,25)

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, 5465);