C₇×#5124 #13056159

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

Aliases

direct product

C₇×#5124

Splittings

Hall	C₃×C₇ ⋊ #430
Sylow	C₃ ⋊ C₇×#430

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×#67
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂×C₃×C₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₇	Q₈
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₂×C₇×M₁₆	D₃
hypocenter	C₃	C₇×#430
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×M₁₆

204 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 118 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)(11,13,14,22,16,23,15,24)(12,18,19,21,17,25,20,26);(12,17)(18,25)(19,20)(21,26);(9,10)(11,12,15,20,16,17,14,19)(13,21,22,25,23,26,24,18)

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, 5448);