C₇×#5837 #13061553

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#5837

Splittings

Fitting	C₃×C₇×2₋¹⁺⁴ ⋊ C₂
Hall	C₃×C₇ ⋊ #577
Sylow	C₃ ⋊ C₇×#577

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂²×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₃×C₇×2₋¹⁺⁴	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₇×2₋¹⁺⁴	D₃
hypocenter	C₃	C₇×#577
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₇×#41 ◁ C₇×2₋¹⁺⁴

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

normal subgroups only

This group has 516 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) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2,3,4,5,6,7)(8,9,10)(11,13,15,20)(12,17,16,23)(14,22,21,19)(18,26,24,25);(12,16)(17,23)(18,24)(25,26);(11,12)(13,17)(14,18)(15,16)(19,25)(20,23)(21,24)(22,26);(9,10)(13,19)(14,21)(17,25)(18,24)(20,22)(23,26)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10842);