C₇×#4915 #13055950

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#4915

Splittings

Fitting	C₃×C₇×SD₃₂ ⋊ C₂
Hall	C₃×C₇ ⋊ #510
Sylow	C₃ ⋊ C₇×#510

Subgroups

	subgroup	G/H
center	C₂×C₇	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³×C₇
Frattini	C₈	C₂²×D₃×C₇
Fitting	C₃×C₇×SD₃₂	C₂
socle	C₂×C₃×C₇	C₂×D₈
prime radical	C₇×#912	C₂
hypercenter	C₇×SD₃₂	D₃
hypocenter	C₃	C₇×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×SD₃₂

164 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 62 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)(12,18)(13,21)(14,25)(15,26)(17,29)(19,36)(22,23)(24,40)(27,41)(28,39)(30,38)(31,42)(32,35)(33,37);(11,13,16,24)(12,17,18,32)(14,21,27,40)(15,28,26,23)(19,35,33,29)(20,38,34,31)(22,41,39,25)(30,36,42,37);(9,10)(11,12)(13,17)(14,19)(15,20)(16,18)(21,35)(22,30)(23,31)(24,32)(25,37)(26,34)(27,33)(28,38)(29,40)(36,41)(39,42)

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5239);