C₃×#90526 #13061177

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

Aliases

direct product

C₃×#90526

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₇ ⋊ #576
Sylow	C₇ ⋊ C₃×#576

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²×C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₇
Fitting	C₃×C₇×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₃×H(5,𝔽₂)	D₇
hypocenter	C₇	C₃×#576
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41 ◁ C₃×H(5,𝔽₂)

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

normal subgroups only

This group has 536 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)(4,5,7,9,10,8,6)(12,17)(14,22)(16,24)(18,19)(21,26)(23,25);(13,20)(16,23)(21,26)(24,25);(11,13)(12,16)(14,21)(15,20)(17,24)(18,25)(19,23)(22,26);(5,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,22)(20,23)(21,24)(25,26)

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, 10466);