C₇×#5203 #13056238

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

Aliases

direct product

C₇×#5203

Splittings

Fitting	C₃×C₇×#137 ⋊ C₂
Hall	C₃×C₇ ⋊ #471
Sylow	C₃ ⋊ C₇×#471

Subgroups

	subgroup	G/H
center	C₂×C₇	#939
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#137	C₂
socle	C₂×C₃×C₇	#121
prime radical	C₇×#949	C₂
hypercenter	C₇×#137	D₃
hypocenter	C₃	C₇×#471
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#137

208 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 74 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)(11,13,16,23)(12,18,19,26)(14,22,15,20)(17,25,24,21);(11,12)(13,21)(14,17)(15,24)(16,19)(18,22)(20,26)(23,25);(9,10)(12,17)(13,20)(14,15)(19,24)(21,25)(22,23)

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, 5527);