C₇×#5836 #13061552

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#5836

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₇ ⋊ #577
Sylow	C₃ ⋊ C₇×#577

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂²×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₃×C₇×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₇×H(5,𝔽₂)	D₃
hypocenter	C₃	C₇×#577
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₇×#41 ◁ C₇×H(5,𝔽₂)

516 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 214 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 — 4 generators on 26 points:

(1,2,3,4,5,6,7)(8,9,10)(11,15,17,14)(12,20,21,26)(13,22,16,23)(18,19,25,24);(11,14,17,15)(12,19,21,24)(13,22,16,23)(18,20,25,26);(11,13,17,16)(12,18,21,25)(14,23,15,22)(19,26,24,20);(9,10)(11,12)(13,18)(14,19)(15,24)(16,25)(17,21)(20,22)(23,26)

Conjugacy classes & character table

273 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10841);