C₃×#89817 #13055478

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#89817

Splittings

Hall	C₃×C₇ ⋊ #485
Sylow	C₇ ⋊ C₃×#485

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂³×C₃×C₇	D₄
hypercenter	C₃×C₂²⋊Q₈	D₇
hypocenter	C₇	C₃×#485
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×C₂²⋊Q₈

192 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 34 points:

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

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, 4767);