C₃×#89825 #13055486

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

Aliases

direct product

C₃×#89825

Splittings

Fitting	C₃×C₇×#123 ⋊ C₂
Hall	C₃×C₇ ⋊ #458
Sylow	C₇ ⋊ C₃×#458

Subgroups

	subgroup	G/H
center	C₂×C₃	#6500
commutator, cosocle	C₂³×C₇	C₂³×C₃
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₃×C₇×#123	C₂
socle	C₂×C₃×C₇	#120
prime radical	C₃×#6548	C₂
hypercenter	C₃×#123	D₇
hypocenter	C₇	C₃×#458
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ C₃×#123

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

normal subgroups only

This group has 320 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)(11,14,15,21)(12,17,18,25)(13,19,20,26)(16,22,23,24);(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,26)(22,25);(5,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,24)(20,23)(21,25)(22,26)

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, 4775);