C₇×#5120 #13056171

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#5120

Splittings

Hall	C₃×C₇ ⋊ #419
Sylow	C₃ ⋊ C₇×#419

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂³×C₃×C₇	D₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂×C₇×M₁₆	D₃
hypocenter	C₃	C₇×#419
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×M₁₆

300 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 126 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)(11,13,14,12)(15,18,19,26,20,27,17,24)(16,22,21,25,23,30,28,29);(15,17,20,19)(16,21,23,28)(18,26,27,24)(22,29,30,25);(9,10)(11,12,14,13)(15,16)(17,21)(18,25)(19,28)(20,23)(22,26)(24,30)(27,29)

Conjugacy classes & character table

294 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5460);