C₇×#5145 #13056180

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

Aliases

direct product

C₇×#5145

Splittings

Fitting	C₃×C₇×#131 ⋊ C₂
Hall	C₃×C₇ ⋊ #444
Sylow	C₃ ⋊ C₇×#444

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×#67
commutator	C₃×C₄	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#131	C₂
socle	C₂×C₃×C₇	C₄×D₄
prime radical	C₇×#950	C₂
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#131	D₃
hypocenter	C₃	C₇×#444
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₇×#131

208 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 94 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)(11,14,15,18,17,25,16,24)(12,13,19,21,20,22,23,26);(11,13)(12,18)(14,23)(15,21)(16,26)(17,22)(19,25)(20,24);(9,10)(11,12)(13,14)(15,19)(16,23)(17,20)(18,21)(22,25)(24,26)

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, 5469);