C₇×#4714 #13055739

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

Aliases

direct product

C₇×#4714

Splittings

Fitting	C₃×C₇×#97 ⋊ C₂
Hall	C₃×C₇ ⋊ #489
Sylow	C₃ ⋊ C₇×#489

Subgroups

	subgroup	G/H
center	C₂²×C₇	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#97	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₇×D₁₂	C₂²
hypercenter	C₇×#97	D₃
hypocenter	C₃	C₇×#489
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#97

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

normal subgroups only

This group has 200 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) = 672

A minimal faithful permutation representation — 3 generators on 26 points:

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

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, 5028);