C₇×#5053 #13056088

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

Aliases

direct product

C₇×#5053

Splittings

Fitting	C₃×C₇×#123 ⋊ C₂
Hall	C₃×C₇ ⋊ #458
Sylow	C₃ ⋊ C₇×#458

Subgroups

	subgroup	G/H
center	C₂×C₇	#937
commutator, cosocle	C₂³×C₃	C₂³×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₃×C₇×#123	C₂
socle	C₂×C₃×C₇	#120
prime radical	C₇×#1004	C₂
hypercenter	C₇×#123	D₃
hypocenter	C₃	C₇×#458
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₇×#123

320 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,14,15,21)(12,17,18,25)(13,19,20,26)(16,22,23,24);(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,26)(22,25);(9,10)(11,12)(13,16)(14,17)(15,18)(19,24)(20,23)(21,25)(22,26)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5377);