C₇×#5237 #13056273

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

Aliases

direct product

C₇×#5237

Splittings

Hall	C₃×C₇ ⋊ #413
Sylow	C₃ ⋊ C₇×#413

Subgroups

	subgroup	G/H
center	C₄×C₇	#217
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting, prime radical	C₂×C₃×C₇×#41	C₂
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂×C₇×#41	D₃
hypocenter	C₃	C₇×#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#41

300 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 134 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)(12,17)(13,19)(14,21)(22,26);(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26);(9,10)(11,12,14,18,15,17,21,24)(13,16,20,22,19,23,25,26)

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, 5562);