C₃×#89834 #13055495

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

Aliases

direct product

C₃×#89834

Splittings

Fitting	C₃×C₇×#124 ⋊ C₂
Hall	C₃×C₇ ⋊ #454
Sylow	C₇ ⋊ C₃×#454

Subgroups

	subgroup	G/H
center	C₂×C₃	#6500
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#124	C₂
socle	C₂×C₃×C₇	#120
hypercenter	C₃×#124	D₇
hypocenter	C₇	C₃×#454
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#124

292 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 78 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)(14,23)(16,24)(18,19)(20,22)(25,26);(11,13,15,21)(12,16,19,26)(14,22,23,20)(17,25,18,24);(5,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,23)(20,24)(21,26)(22,25)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4784);