C₃×#89512 #13055173

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃×#89512

Splittings

Fitting	C₃×C₇×#99 ⋊ C₂
Hall	C₃×C₇ ⋊ #410
Sylow	C₇ ⋊ C₃×#410

Subgroups

	subgroup	G/H
center	C₂×C₃	D₇×C₂²⋊C₄
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₃×C₇×#99	C₂
socle	C₂×C₃×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₃×#1159	C₂
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₃×#99	D₇
hypocenter	C₇	C₃×#410
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ C₃×#99

316 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 3 generators on 26 points:

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4462);