Copied to
clipboard

G = C37⋊C3order 111 = 3·37

The semidirect product of C37 and C3 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C37⋊C3, SmallGroup(111,1)

Series: Derived Chief Lower central Upper central

C1C37 — C37⋊C3
C1C37 — C37⋊C3
C37 — C37⋊C3
C1

Generators and relations for C37⋊C3
 G = < a,b | a37=b3=1, bab-1=a10 >

37C3

Character table of C37⋊C3

 class 13A3B37A37B37C37D37E37F37G37H37I37J37K37L
 size 13737333333333333
ρ1111111111111111    trivial
ρ21ζ32ζ3111111111111    linear of order 3
ρ31ζ3ζ32111111111111    linear of order 3
ρ4300ζ37343733377ζ373237243718ζ373137293714ζ3730374373ζ372837253721ζ3726371037ζ3723378376ζ37203715372ζ37193713375ζ37163712379ζ373637273711ζ373537223717    complex faithful
ρ5300ζ37203715372ζ372837253721ζ3730374373ζ373537223717ζ3723378376ζ373237243718ζ37343733377ζ373637273711ζ37163712379ζ373137293714ζ37193713375ζ3726371037    complex faithful
ρ6300ζ373637273711ζ3723378376ζ373537223717ζ3726371037ζ37343733377ζ372837253721ζ37203715372ζ37193713375ζ373137293714ζ3730374373ζ37163712379ζ373237243718    complex faithful
ρ7300ζ37193713375ζ37343733377ζ3726371037ζ373237243718ζ37203715372ζ3723378376ζ373637273711ζ37163712379ζ3730374373ζ373537223717ζ373137293714ζ372837253721    complex faithful
ρ8300ζ3726371037ζ373137293714ζ37203715372ζ373637273711ζ3730374373ζ37163712379ζ373537223717ζ373237243718ζ3723378376ζ37343733377ζ372837253721ζ37193713375    complex faithful
ρ9300ζ3730374373ζ37193713375ζ3723378376ζ37343733377ζ37163712379ζ373637273711ζ373137293714ζ373537223717ζ373237243718ζ372837253721ζ3726371037ζ37203715372    complex faithful
ρ10300ζ372837253721ζ373537223717ζ37193713375ζ37163712379ζ3726371037ζ3730374373ζ373237243718ζ3723378376ζ37203715372ζ373637273711ζ37343733377ζ373137293714    complex faithful
ρ11300ζ373237243718ζ3730374373ζ373637273711ζ37193713375ζ373537223717ζ373137293714ζ3726371037ζ372837253721ζ37343733377ζ37203715372ζ3723378376ζ37163712379    complex faithful
ρ12300ζ37163712379ζ37203715372ζ373237243718ζ372837253721ζ373637273711ζ37343733377ζ37193713375ζ373137293714ζ373537223717ζ3726371037ζ3730374373ζ3723378376    complex faithful
ρ13300ζ3723378376ζ3726371037ζ37163712379ζ373137293714ζ373237243718ζ373537223717ζ372837253721ζ37343733377ζ373637273711ζ37193713375ζ37203715372ζ3730374373    complex faithful
ρ14300ζ373137293714ζ373637273711ζ372837253721ζ3723378376ζ37193713375ζ37203715372ζ37163712379ζ3730374373ζ3726371037ζ373237243718ζ373537223717ζ37343733377    complex faithful
ρ15300ζ373537223717ζ37163712379ζ37343733377ζ37203715372ζ373137293714ζ37193713375ζ3730374373ζ3726371037ζ372837253721ζ3723378376ζ373237243718ζ373637273711    complex faithful

Smallest permutation representation of C37⋊C3
On 37 points: primitive
Generators in S37
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37)
(2 27 11)(3 16 21)(4 5 31)(6 20 14)(7 9 24)(8 35 34)(10 13 17)(12 28 37)(15 32 30)(18 36 23)(19 25 33)(22 29 26)

G:=sub<Sym(37)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)], [(2,27,11),(3,16,21),(4,5,31),(6,20,14),(7,9,24),(8,35,34),(10,13,17),(12,28,37),(15,32,30),(18,36,23),(19,25,33),(22,29,26)])

C37⋊C3 is a maximal subgroup of   C37⋊C6  C37⋊C9  C37⋊A4
C37⋊C3 is a maximal quotient of   C372C9  C37⋊A4

Matrix representation of C37⋊C3 in GL3(𝔽223) generated by

5210
11201
184193141
,
621795
66206159
5514178
G:=sub<GL(3,GF(223))| [52,112,184,1,0,193,0,1,141],[62,66,55,179,206,14,5,159,178] >;

C37⋊C3 in GAP, Magma, Sage, TeX

C_{37}\rtimes C_3
% in TeX

G:=Group("C37:C3");
// GroupNames label

G:=SmallGroup(111,1);
// by ID

G=gap.SmallGroup(111,1);
# by ID

G:=PCGroup([2,-3,-37,313]);
// Polycyclic

G:=Group<a,b|a^37=b^3=1,b*a*b^-1=a^10>;
// generators/relations

Export

Subgroup lattice of C37⋊C3 in TeX
Character table of C37⋊C3 in TeX

׿
×
𝔽