C₇×#5835 #13061551

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

Aliases

direct product

C₇×#5835

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₇ ⋊ #576
Sylow	C₃ ⋊ C₇×#576

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×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₇×#41 ◁ C₇×H(5,𝔽₂)

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

normal subgroups only

This group has 536 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)(12,17)(14,22)(16,24)(18,19)(21,26)(23,25);(13,20)(16,23)(21,26)(24,25);(11,13)(12,16)(14,21)(15,20)(17,24)(18,25)(19,23)(22,26);(9,10)(11,12)(13,16)(14,17)(15,19)(18,22)(20,23)(21,24)(25,26)

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, 10840);