C₃×#89569 #13055229

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#89569

Splittings

Hall	C₃×C₇ ⋊ #491
Sylow	C₇ ⋊ C₃×#491

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×#103	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₃×#103	D₇
hypocenter	C₇	C₃×#491
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#103

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

normal subgroups only

This group has 180 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) = 34
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3)(4,5,7,9,10,8,6)(11,13,14,18)(12,16,15,17)(20,25)(21,27)(22,30)(24,31)(28,29)(32,34);(12,15)(16,17)(19,21,23,29)(20,24,25,32)(22,27,30,28)(26,34,33,31);(5,6)(7,8)(9,10)(11,12,14,15)(13,17,18,16)(19,20)(21,24)(22,26)(23,25)(27,34)(28,31)(29,32)(30,33)

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, 4518);