C₇×#5062 #13056097

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

Aliases

direct product

C₇×#5062

Splittings

Fitting	C₃×C₇×#124 ⋊ C₂
Hall	C₃×C₇ ⋊ #454
Sylow	C₃ ⋊ C₇×#454

Subgroups

	subgroup	G/H
center	C₂×C₇	#937
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#124	C₂
socle	C₂×C₃×C₇	#120
hypercenter	C₇×#124	D₃
hypocenter	C₃	C₇×#454
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#124

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

normal subgroups only

This group has 292 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)(12,17)(14,23)(16,24)(18,19)(20,22)(25,26);(11,13,15,21)(12,16,19,26)(14,22,23,20)(17,25,18,24);(9,10)(11,12)(13,16)(14,17)(15,19)(18,23)(20,24)(21,26)(22,25)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5386);