C₇×#5125 #13056160

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

Aliases

direct product

C₇×#5125

Splittings

Hall	C₃×C₇ ⋊ #430
Sylow	C₃ ⋊ C₇×#430

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×D₁₂
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 134 normal subgroups are shown.

normal subgroups only

This group has 204 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)(11,13,15,23,16,21,14,22)(12,17,18,20,19,25,24,26);(13,21)(17,25)(20,26)(22,23);(9,10)(11,12,14,18,16,19,15,24)(13,20,22,25,21,26,23,17)

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, 5449);