C₇×#5006 #13056038

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

Aliases

direct product

C₇×#5006

Splittings

Fitting	C₃×C₇×#117 ⋊ C₂
Hall	C₃×C₇ ⋊ #410
Sylow	C₃ ⋊ C₇×#410

Subgroups

	subgroup	G/H
center	C₄×C₇	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₃×C₇×#117	C₂
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₇×#235	C₂
cosocle	C₂³×C₃	C₂³×C₇
hypercenter	C₇×#117	D₃
hypocenter	C₃	C₇×#410
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#117

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

normal subgroups only

This group has 316 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)(11,14)(12,17,18,21)(13,19)(15,22)(16,23,24,25)(20,26);(11,13,15,20)(12,16,18,24)(14,19,22,26)(17,23,21,25);(9,10)(11,12)(13,16)(14,21)(15,18)(17,22)(19,25)(20,24)(23,26)

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, 5327);