C₇×#4823 #13055858

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4823

Splittings

Fitting	C₃×C₇×#104 ⋊ C₂
Hall	C₃×C₇ ⋊ #444
Sylow	C₃ ⋊ C₇×#444

Subgroups

	subgroup	G/H
center	C₄×C₇	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#104	C₂
socle	C₂×C₃×C₇	C₄×D₄
prime radical	C₇×#950	C₂
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#104	D₃
hypocenter	C₃	C₇×#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#104

208 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 90 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)(11,14,15,24,16,23,25,13)(12,18,19,26,21,17,20,22);(11,13)(12,17)(14,15)(16,24)(18,21)(19,22)(20,26)(23,25);(9,10)(11,12)(13,22)(14,18)(15,19)(16,21)(17,23)(20,25)(24,26)

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, 5147);