C₃×#90529 #13061180

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#90529

Splittings

Fitting	C₃×C₇×2₋¹⁺⁴ ⋊ C₂
Hall	C₃×C₇ ⋊ #578
Sylow	C₇ ⋊ C₃×#578

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²×C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₇
Fitting	C₃×C₇×2₋¹⁺⁴	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₃×2₋¹⁺⁴	D₇
hypocenter	C₇	C₃×#578
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41 ◁ C₃×2₋¹⁺⁴

496 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 214 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 — 4 generators on 42 points:

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

Conjugacy classes & character table

219 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10469);