C₇×#4740 #13055775

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

Aliases

direct product

C₇×#4740

Splittings

Fitting	C₃×C₇×#99 ⋊ C₂
Hall	C₃×C₇ ⋊ #410
Sylow	C₃ ⋊ C₇×#410

Subgroups

	subgroup	G/H
center	C₂×C₇	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₃×C₇×#99	C₂
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₇×#237	C₂
cosocle	C₂³×C₃	C₂³×C₇
hypercenter	C₇×#99	D₃
hypocenter	C₃	C₇×#410
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₇×#99

316 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 102 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)(12,17)(13,19,20,21)(15,22)(16,23,24,25)(18,26);(11,13)(12,16)(14,21)(15,20)(17,25)(18,24)(19,22)(23,26);(9,10)(11,12,15,18)(13,16,20,24)(14,17,22,26)(19,23,21,25)

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5064);