C₇×#5063 #13056099

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#5063

Splittings

Hall	C₃×C₇ ⋊ #479
Sylow	C₃ ⋊ C₇×#479

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#125	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₇×#125	D₃
hypocenter	C₃	C₇×#479
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#125

180 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5388);