C₃×#89520 #13055181

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

Aliases

direct product

C₃×#89520

Splittings

Fitting	C₃×C₇×#100 ⋊ C₂
Hall	C₃×C₇ ⋊ #471
Sylow	C₇ ⋊ C₃×#471

Subgroups

	subgroup	G/H
center	C₂×C₃	#6447
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#100	C₂
socle	C₂×C₃×C₇	#121
prime radical	C₃×#6495	C₂
hypercenter	C₃×#100	D₇
hypocenter	C₇	C₃×#471
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#100

208 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 70 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)(13,19)(14,21)(16,22)(18,24)(23,26);(11,13,14,20,15,19,21,25)(12,16,17,22,18,23,24,26);(5,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,23)(20,22)(21,24)(25,26)

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, 4470);