C₇×#5000 #13056035

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

Aliases

direct product

C₇×#5000

Splittings

Hall	C₃×C₇ ⋊ #433
Sylow	C₃ ⋊ C₇×#433

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×#117	D₃
hypocenter	C₃	C₇×#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#117

192 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 34
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3,4,5,6,7)(8,9,10)(12,16)(15,18)(21,27)(24,30)(28,33)(31,34);(11,13,14,17)(12,15,16,18)(19,21)(20,24)(22,28)(23,27)(25,31)(26,30)(29,33)(32,34);(9,10)(11,12)(13,15)(14,16)(17,18)(19,20,22,25,23,26,29,32)(21,24,28,31,27,30,33,34)

Conjugacy classes & character table

336 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5324);