C₃×#89521 #13055182

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

Aliases

direct product

C₃×#89521

Splittings

Fitting	C₃×C₇×#100 ⋊ C₂
Hall	C₃×C₇ ⋊ #472
Sylow	C₇ ⋊ C₃×#472

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
hypercenter	C₃×#100	D₇
hypocenter	C₇	C₃×#472
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#100

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

normal subgroups only

This group has 216 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)(13,20)(14,22)(16,23)(18,19)(24,25);(11,13,14,21,15,20,22,26)(12,16,18,24,19,25,17,23);(5,6)(7,8)(9,10)(11,12)(13,16)(14,18)(15,19)(17,22)(20,25)(21,24)(23,26)

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, 4471);