C₇×#4742 #13055777

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

Aliases

direct product

C₇×#4742

Splittings

Fitting	C₃×C₇×#99 ⋊ C₂
Hall	C₃×C₇ ⋊ #458
Sylow	C₃ ⋊ C₇×#458

Subgroups

	subgroup	G/H
center	C₂×C₇	#882
commutator, cosocle	C₂³×C₃	C₂³×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₃×C₇×#99	C₂
socle	C₂×C₃×C₇	#120
prime radical	C₇×#1004	C₂
hypercenter	C₇×#99	D₃
hypocenter	C₃	C₇×#458
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₇×#99

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)(11,14,15,22)(12,17,18,25)(13,19)(16,23)(20,21)(24,26);(11,13)(12,16)(14,21)(15,20)(17,23)(18,24)(19,22)(25,26);(9,10)(11,12)(13,16)(14,17)(15,18)(19,26)(20,24)(21,23)(22,25)

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, 5066);