C₃×#89972 #13055632

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#89972

Splittings

Hall	C₃×C₇ ⋊ #471
Sylow	C₇ ⋊ C₃×#471

Subgroups

	subgroup	G/H
center	C₂×C₃	#6502
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting, prime radical	C₃×C₇×AGL(1,ℤ/8)	C₂
socle	C₂×C₃×C₇	#121
hypercenter	C₃×AGL(1,ℤ/8)	D₇
hypocenter	C₇	C₃×#471
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×AGL(1,ℤ/8)

208 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 74 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)(4,5,7,9,10,8,6)(12,17)(13,21)(14,24)(18,19)(20,25)(22,23);(11,13)(12,16)(14,21)(15,23)(17,20)(18,25)(19,26)(22,24);(5,6)(7,8)(9,10)(11,12,15,19)(13,20,23,25)(14,17,24,18)(16,21,26,22)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4921);