C₇×#4907 #13055942

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4907

Splittings

Fitting	C₃×C₇×M₃₂ ⋊ C₂
Hall	C₃×C₇ ⋊ #504
Sylow	C₃ ⋊ C₇×#504

Subgroups

	subgroup	G/H
center	C₇×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₇×C₈
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×M₃₂	C₂
socle	C₂×C₃×C₇	C₂²×C₈
prime radical	C₇×#235	C₄
cosocle	C₃×C₈	C₂³×C₇
hypercenter	C₇×M₃₂	D₃
hypocenter	C₃	C₇×#504
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₇×M₃₂

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

normal subgroups only

This group has 168 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) = 42
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3,4,5,6,7)(8,9,10)(11,13,14,22,15,23,24,35,16,21,25,33,26,34,36,41)(12,18,19,30,20,31,32,40,17,27,28,37,29,38,39,42);(12,17)(13,21)(19,28)(20,29)(22,33)(23,34)(32,39)(35,41);(9,10)(11,12)(13,18)(14,19)(15,20)(16,17)(21,27)(22,30)(23,31)(24,32)(25,28)(26,29)(33,37)(34,38)(35,40)(36,39)(41,42)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5231);