C₇×#5819 #13061535

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#5819

Splittings

Hall	C₃×C₇ ⋊ #568
Sylow	C₃ ⋊ C₇×#568

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting, prime radical	C₂×C₃×C₇×#41	C₂
socle	C₂×C₃×C₇	C₂³×C₄
cosocle	C₃×C₄	C₂⁴×C₇
hypercenter	C₂×C₇×#41	D₃
hypocenter	C₃	C₇×#568
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×#41

516 conjugacy classes of subgroups; only the 374 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 374 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)(16,22)(18,24)(23,26);(13,19)(16,22)(20,25)(23,26);(11,13)(12,16)(14,20)(15,19)(17,22)(18,23)(21,25)(24,26);(9,10)(11,12,14,18,15,17,21,24)(13,16,20,23,19,22,25,26)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10824);