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G = C3×D13order 78 = 2·3·13

Direct product of C3 and D13

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×D13, C392C2, C133C6, SmallGroup(78,4)

Series: Derived Chief Lower central Upper central

C1C13 — C3×D13
C1C13C39 — C3×D13
C13 — C3×D13
C1C3

Generators and relations for C3×D13
 G = < a,b,c | a3=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C6

Character table of C3×D13

 class 123A3B6A6B13A13B13C13D13E13F39A39B39C39D39E39F39G39H39I39J39K39L
 size 113111313222222222222222222
ρ1111111111111111111111111    trivial
ρ21-111-1-1111111111111111111    linear of order 2
ρ311ζ32ζ3ζ32ζ3111111ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ41-1ζ3ζ32ζ65ζ6111111ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 6
ρ51-1ζ32ζ3ζ6ζ65111111ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 6
ρ611ζ3ζ32ζ3ζ32111111ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ7202200ζ139134ζ138135ζ1310133ζ1311132ζ137136ζ131213ζ131213ζ139134ζ138135ζ138135ζ1310133ζ1311132ζ137136ζ131213ζ139134ζ1310133ζ1311132ζ137136    orthogonal lifted from D13
ρ8202200ζ131213ζ1311132ζ139134ζ137136ζ138135ζ1310133ζ1310133ζ131213ζ1311132ζ1311132ζ139134ζ137136ζ138135ζ1310133ζ131213ζ139134ζ137136ζ138135    orthogonal lifted from D13
ρ9202200ζ138135ζ1310133ζ137136ζ139134ζ131213ζ1311132ζ1311132ζ138135ζ1310133ζ1310133ζ137136ζ139134ζ131213ζ1311132ζ138135ζ137136ζ139134ζ131213    orthogonal lifted from D13
ρ10202200ζ137136ζ131213ζ1311132ζ1310133ζ139134ζ138135ζ138135ζ137136ζ131213ζ131213ζ1311132ζ1310133ζ139134ζ138135ζ137136ζ1311132ζ1310133ζ139134    orthogonal lifted from D13
ρ11202200ζ1310133ζ137136ζ131213ζ138135ζ1311132ζ139134ζ139134ζ1310133ζ137136ζ137136ζ131213ζ138135ζ1311132ζ139134ζ1310133ζ131213ζ138135ζ1311132    orthogonal lifted from D13
ρ12202200ζ1311132ζ139134ζ138135ζ131213ζ1310133ζ137136ζ137136ζ1311132ζ139134ζ139134ζ138135ζ131213ζ1310133ζ137136ζ1311132ζ138135ζ131213ζ1310133    orthogonal lifted from D13
ρ1320-1+-3-1--300ζ1310133ζ137136ζ131213ζ138135ζ1311132ζ139134ζ32ζ13932ζ134ζ32ζ131032ζ133ζ32ζ13732ζ136ζ3ζ1373ζ136ζ3ζ13123ζ13ζ3ζ1383ζ135ζ3ζ13113ζ132ζ3ζ1393ζ134ζ3ζ13103ζ133ζ32ζ131232ζ13ζ32ζ13832ζ135ζ32ζ131132ζ132    complex faithful
ρ1420-1--3-1+-300ζ139134ζ138135ζ1310133ζ1311132ζ137136ζ131213ζ3ζ13123ζ13ζ3ζ1393ζ134ζ3ζ1383ζ135ζ32ζ13832ζ135ζ32ζ131032ζ133ζ32ζ131132ζ132ζ32ζ13732ζ136ζ32ζ131232ζ13ζ32ζ13932ζ134ζ3ζ13103ζ133ζ3ζ13113ζ132ζ3ζ1373ζ136    complex faithful
ρ1520-1+-3-1--300ζ137136ζ131213ζ1311132ζ1310133ζ139134ζ138135ζ32ζ13832ζ135ζ32ζ13732ζ136ζ32ζ131232ζ13ζ3ζ13123ζ13ζ3ζ13113ζ132ζ3ζ13103ζ133ζ3ζ1393ζ134ζ3ζ1383ζ135ζ3ζ1373ζ136ζ32ζ131132ζ132ζ32ζ131032ζ133ζ32ζ13932ζ134    complex faithful
ρ1620-1+-3-1--300ζ139134ζ138135ζ1310133ζ1311132ζ137136ζ131213ζ32ζ131232ζ13ζ32ζ13932ζ134ζ32ζ13832ζ135ζ3ζ1383ζ135ζ3ζ13103ζ133ζ3ζ13113ζ132ζ3ζ1373ζ136ζ3ζ13123ζ13ζ3ζ1393ζ134ζ32ζ131032ζ133ζ32ζ131132ζ132ζ32ζ13732ζ136    complex faithful
ρ1720-1--3-1+-300ζ1310133ζ137136ζ131213ζ138135ζ1311132ζ139134ζ3ζ1393ζ134ζ3ζ13103ζ133ζ3ζ1373ζ136ζ32ζ13732ζ136ζ32ζ131232ζ13ζ32ζ13832ζ135ζ32ζ131132ζ132ζ32ζ13932ζ134ζ32ζ131032ζ133ζ3ζ13123ζ13ζ3ζ1383ζ135ζ3ζ13113ζ132    complex faithful
ρ1820-1--3-1+-300ζ137136ζ131213ζ1311132ζ1310133ζ139134ζ138135ζ3ζ1383ζ135ζ3ζ1373ζ136ζ3ζ13123ζ13ζ32ζ131232ζ13ζ32ζ131132ζ132ζ32ζ131032ζ133ζ32ζ13932ζ134ζ32ζ13832ζ135ζ32ζ13732ζ136ζ3ζ13113ζ132ζ3ζ13103ζ133ζ3ζ1393ζ134    complex faithful
ρ1920-1+-3-1--300ζ138135ζ1310133ζ137136ζ139134ζ131213ζ1311132ζ32ζ131132ζ132ζ32ζ13832ζ135ζ32ζ131032ζ133ζ3ζ13103ζ133ζ3ζ1373ζ136ζ3ζ1393ζ134ζ3ζ13123ζ13ζ3ζ13113ζ132ζ3ζ1383ζ135ζ32ζ13732ζ136ζ32ζ13932ζ134ζ32ζ131232ζ13    complex faithful
ρ2020-1--3-1+-300ζ138135ζ1310133ζ137136ζ139134ζ131213ζ1311132ζ3ζ13113ζ132ζ3ζ1383ζ135ζ3ζ13103ζ133ζ32ζ131032ζ133ζ32ζ13732ζ136ζ32ζ13932ζ134ζ32ζ131232ζ13ζ32ζ131132ζ132ζ32ζ13832ζ135ζ3ζ1373ζ136ζ3ζ1393ζ134ζ3ζ13123ζ13    complex faithful
ρ2120-1--3-1+-300ζ1311132ζ139134ζ138135ζ131213ζ1310133ζ137136ζ3ζ1373ζ136ζ3ζ13113ζ132ζ3ζ1393ζ134ζ32ζ13932ζ134ζ32ζ13832ζ135ζ32ζ131232ζ13ζ32ζ131032ζ133ζ32ζ13732ζ136ζ32ζ131132ζ132ζ3ζ1383ζ135ζ3ζ13123ζ13ζ3ζ13103ζ133    complex faithful
ρ2220-1+-3-1--300ζ1311132ζ139134ζ138135ζ131213ζ1310133ζ137136ζ32ζ13732ζ136ζ32ζ131132ζ132ζ32ζ13932ζ134ζ3ζ1393ζ134ζ3ζ1383ζ135ζ3ζ13123ζ13ζ3ζ13103ζ133ζ3ζ1373ζ136ζ3ζ13113ζ132ζ32ζ13832ζ135ζ32ζ131232ζ13ζ32ζ131032ζ133    complex faithful
ρ2320-1+-3-1--300ζ131213ζ1311132ζ139134ζ137136ζ138135ζ1310133ζ32ζ131032ζ133ζ32ζ131232ζ13ζ32ζ131132ζ132ζ3ζ13113ζ132ζ3ζ1393ζ134ζ3ζ1373ζ136ζ3ζ1383ζ135ζ3ζ13103ζ133ζ3ζ13123ζ13ζ32ζ13932ζ134ζ32ζ13732ζ136ζ32ζ13832ζ135    complex faithful
ρ2420-1--3-1+-300ζ131213ζ1311132ζ139134ζ137136ζ138135ζ1310133ζ3ζ13103ζ133ζ3ζ13123ζ13ζ3ζ13113ζ132ζ32ζ131132ζ132ζ32ζ13932ζ134ζ32ζ13732ζ136ζ32ζ13832ζ135ζ32ζ131032ζ133ζ32ζ131232ζ13ζ3ζ1393ζ134ζ3ζ1373ζ136ζ3ζ1383ζ135    complex faithful

Smallest permutation representation of C3×D13
On 39 points
Generators in S39
(1 38 25)(2 39 26)(3 27 14)(4 28 15)(5 29 16)(6 30 17)(7 31 18)(8 32 19)(9 33 20)(10 34 21)(11 35 22)(12 36 23)(13 37 24)
(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 38 39)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 35)(28 34)(29 33)(30 32)(36 39)(37 38)

G:=sub<Sym(39)| (1,38,25)(2,39,26)(3,27,14)(4,28,15)(5,29,16)(6,30,17)(7,31,18)(8,32,19)(9,33,20)(10,34,21)(11,35,22)(12,36,23)(13,37,24), (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)(27,28,29,30,31,32,33,34,35,36,37,38,39), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,35)(28,34)(29,33)(30,32)(36,39)(37,38)>;

G:=Group( (1,38,25)(2,39,26)(3,27,14)(4,28,15)(5,29,16)(6,30,17)(7,31,18)(8,32,19)(9,33,20)(10,34,21)(11,35,22)(12,36,23)(13,37,24), (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)(27,28,29,30,31,32,33,34,35,36,37,38,39), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,35)(28,34)(29,33)(30,32)(36,39)(37,38) );

G=PermutationGroup([(1,38,25),(2,39,26),(3,27,14),(4,28,15),(5,29,16),(6,30,17),(7,31,18),(8,32,19),(9,33,20),(10,34,21),(11,35,22),(12,36,23),(13,37,24)], [(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),(27,28,29,30,31,32,33,34,35,36,37,38,39)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,35),(28,34),(29,33),(30,32),(36,39),(37,38)])

C3×D13 is a maximal subgroup of   C39⋊C4  C13⋊C18

Matrix representation of C3×D13 in GL2(𝔽79) generated by

230
023
,
401
780
,
01
10
G:=sub<GL(2,GF(79))| [23,0,0,23],[40,78,1,0],[0,1,1,0] >;

C3×D13 in GAP, Magma, Sage, TeX

C_3\times D_{13}
% in TeX

G:=Group("C3xD13");
// GroupNames label

G:=SmallGroup(78,4);
// by ID

G=gap.SmallGroup(78,4);
# by ID

G:=PCGroup([3,-2,-3,-13,650]);
// Polycyclic

G:=Group<a,b,c|a^3=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D13 in TeX
Character table of C3×D13 in TeX

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