C₇×#5003 #13056039

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

Aliases

direct product

C₇×#5003

Splittings

Fitting	C₃×C₇×#117 ⋊ C₂
Hall	C₃×C₇ ⋊ #482
Sylow	C₃ ⋊ C₇×#482

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×#117	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₇×#930	C₂
hypercenter	C₇×#117	D₃
hypocenter	C₃	C₇×#482
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#117

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

normal subgroups only

This group has 212 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)(11,13,14,17)(12,16,15,18)(19,21,23,25)(20,24,22,26);(12,15)(16,18)(19,20,23,22)(21,24,25,26);(9,10)(11,12)(13,16)(14,15)(17,18)(20,22)(21,24)(25,26)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5328);