C₇×#5051 #13056081

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

Aliases

direct product

C₇×#5051

Splittings

Hall	C₃×C₇ ⋊ #467
Sylow	C₃ ⋊ C₇×#467

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂³×C₃×C₇	D₄
hypercenter	C₇×C₂²⋊Q₈	D₃
hypocenter	C₃	C₇×#467
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×C₂²⋊Q₈

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

normal subgroups only

This group has 228 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) = 30
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3,4,5,6,7)(8,9,10)(12,14)(15,17,19,25)(16,20,22,29)(18,27,28,24)(21,30,26,23);(11,12)(13,14);(9,10)(12,14)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24)

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, 5370);