C₃×#90363 #13061015

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

Aliases

direct product

C₃×#90363

Splittings

Hall	C₃×C₇ ⋊ #563
Sylow	C₇ ⋊ C₃×#563

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#124	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	C₃×#124	D₇
hypocenter	C₇	C₃×#563
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

392 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 34
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10304);