C₇×#4768 #13055802

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#4768

Splittings

Hall	C₃×C₇ ⋊ #488
Sylow	C₃ ⋊ C₇×#488

Subgroups

	subgroup	G/H
center	C₂²×C₇	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#102	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	C₇×#102	D₃
hypocenter	C₃	C₇×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#102

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

normal subgroups only

This group has 192 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) = 672

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

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5091);