C₇×#5945 #13062308

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₇ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₇×#5945

Splittings

Fitting, Hall	C₇×#585 ⋊ C₃
Sylow	#585 ⋊ C₃×C₇

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂²×A₄
commutator, cosocle, hypocenter	Q₈	C₂³×C₃×C₇
Frattini	C₂	C₂³×C₇×A₄
Fitting	C₇×#585	C₃
socle	C₂×C₇	C₂³×A₄
hypercenter	C₇×#41	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₄×C₇ ◁ C₇×#41

390 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(9,15)(11,12,21)(14,25)(16,29,30,28,17,31)(18,19,32)(22,24,23)(26,37,38,36,27,39)(33,35,34);(8,32)(9,38)(10,23)(11,35)(12,18)(13,34)(14,31)(15,39)(16,37)(17,26)(19,22)(20,21)(24,33)(25,30)(27,28)(29,36);(8,9)(10,14)(11,16)(12,17)(13,15)(18,26)(19,27)(20,25)(21,30)(22,28)(23,31)(24,29)(32,38)(33,36)(34,39)(35,37)

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11597);