C₇×#5205 #13056240

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

Aliases

direct product

C₇×#5205

Splittings

Fitting	C₃×C₇×#137 ⋊ C₂
Hall	C₃×C₇ ⋊ #456
Sylow	C₃ ⋊ C₇×#456

Subgroups

	subgroup	G/H
center	C₂×C₇	#937
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#137	C₂
socle	C₂×C₃×C₇	#120
prime radical	C₇×#1010	C₂
hypercenter	C₇×#137	D₃
hypocenter	C₃	C₇×#456
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#137

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

normal subgroups only

This group has 284 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)(11,14,17,18)(12,19,21,26)(13,22,23,20)(15,25,16,24);(11,13,17,23)(12,18,21,14)(15,19,16,26)(20,25,22,24);(9,10)(11,12)(13,18)(14,23)(15,22)(16,20)(17,21)(19,24)(25,26)

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, 5529);