C₇×#5748 #13061464

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

Aliases

direct product

C₇×#5748

Splittings

Fitting	C₂×C₃×C₇×M₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ #575
Sylow	C₃ ⋊ C₇×#575

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₂×C₇×M₁₆	D₃
hypocenter	C₃	C₇×#575
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×M₁₆

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

normal subgroups only

This group has 524 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)(11,14,15,23,16,22,13,20)(12,18,17,21,19,25,24,26);(11,13,16,15)(12,17,19,24)(14,20,22,23)(18,21,25,26);(14,22)(18,25)(20,23)(21,26);(9,10)(11,12)(13,17)(14,21)(15,24)(16,19)(18,23)(20,25)(22,26)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10753);