C₇×#4745 #13055780

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

Aliases

direct product

C₇×#4745

Splittings

Fitting	C₃×C₇×#100 ⋊ C₂
Hall	C₃×C₇ ⋊ #413
Sylow	C₃ ⋊ C₇×#413

Subgroups

	subgroup	G/H
center	C₂×C₇	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#100	C₂
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₇×#237	C₂
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#100	D₃
hypocenter	C₃	C₇×#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#100

300 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 102 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)(11,13,14,20,15,19,21,25)(12,17,18,24,16,22,23,26);(12,16)(13,19)(14,21)(24,26);(9,10)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5069);