C₇×#5242 #13056278

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

Aliases

direct product

C₇×#5242

Splittings

Hall	C₃×C₇ ⋊ #455
Sylow	C₃ ⋊ C₇×#455

Subgroups

	subgroup	G/H
center	C₂×C₇	#953
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting, prime radical	C₃×C₇×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₇	#120
hypercenter	C₇×H(5,𝔽₂)	D₃
hypocenter	C₃	C₇×#455
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×H(5,𝔽₂)

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

normal subgroups only

This group has 304 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) = 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)(14,24)(17,22)(19,26)(21,23);(11,13)(12,16)(14,23)(15,20)(17,26)(18,25)(19,22)(21,24);(9,10)(11,12,15,18)(13,19,20,26)(14,22,24,17)(16,23,25,21)

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5567);