C₇×#4749 #13055784

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4749

Splittings

Fitting	C₃×C₇×#100 ⋊ C₂
Hall	C₃×C₇ ⋊ #472
Sylow	C₃ ⋊ C₇×#472

Subgroups

	subgroup	G/H
center	C₂×C₇	#884
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#100	C₂
socle	C₂×C₃×C₇	#121
hypercenter	C₇×#100	D₃
hypocenter	C₃	C₇×#472
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#100

216 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 70 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:

(1,2,3,4,5,6,7)(8,9,10)(12,17)(13,20)(14,22)(16,23)(18,19)(24,25);(11,13,14,21,15,20,22,26)(12,16,18,24,19,25,17,23);(9,10)(11,12)(13,16)(14,18)(15,19)(17,22)(20,25)(21,24)(23,26)

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5073);