C₇×#4825 #13055860

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

Aliases

direct product

C₇×#4825

Splittings

Fitting	C₃×C₇×#104 ⋊ C₂
Hall	C₃×C₇ ⋊ #455
Sylow	C₃ ⋊ C₇×#455

Subgroups

	subgroup	G/H
center	C₂×C₇	#882
commutator, cosocle	C₂×C₃×C₄	C₂³×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#104	C₂
socle	C₂×C₃×C₇	#120
prime radical	C₇×#1009	C₂
hypercenter	C₇×#104	D₃
hypocenter	C₃	C₇×#455
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#104

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

normal subgroups only

This group has 304 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) = 336

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

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5149);