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G = C13⋊D4order 104 = 23·13

The semidirect product of C13 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C132D4, C22⋊D13, D262C2, Dic13⋊C2, C2.5D26, C26.5C22, (C2×C26)⋊2C2, SmallGroup(104,8)

Series: Derived Chief Lower central Upper central

C1C26 — C13⋊D4
C1C13C26D26 — C13⋊D4
C13C26 — C13⋊D4
C1C2C22

Generators and relations for C13⋊D4
 G = < a,b,c | a13=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
26C2
13C4
13C22
2D13
2C26
13D4

Character table of C13⋊D4

 class 12A2B2C413A13B13C13D13E13F26A26B26C26D26E26F26G26H26I26J26K26L26M26N26O26P26Q26R
 size 1122626222222222222222222222222
ρ111111111111111111111111111111    trivial
ρ211-1-111111111111-1-1-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ3111-1-1111111111111111111111111    linear of order 2
ρ411-11-11111111111-1-1-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ52-2000222222-2-2-2-2000000000000-2-2    orthogonal lifted from D4
ρ622200ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ1311132ζ139134ζ1310133ζ137136ζ137136ζ131213ζ138135ζ1311132ζ139134ζ1310133ζ1310133ζ139134ζ1311132ζ138135ζ131213ζ137136ζ131213ζ138135    orthogonal lifted from D13
ρ722200ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ131213ζ137136ζ131213ζ139134ζ138135ζ138135ζ1310133ζ1311132ζ137136ζ131213ζ139134ζ139134ζ131213ζ137136ζ1311132ζ1310133ζ138135ζ1310133ζ1311132    orthogonal lifted from D13
ρ822-200ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ138135ζ139134ζ138135ζ137136ζ1312131312131311132131013313913413813513713613713613813513913413101331311132131213ζ1311132ζ1310133    orthogonal lifted from D26
ρ922-200ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ1311132ζ139134ζ1310133ζ1371361371361312131381351311132139134131013313101331391341311132138135131213137136ζ131213ζ138135    orthogonal lifted from D26
ρ1022200ζ139134ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ131213ζ1311132ζ138135ζ1310133ζ1310133ζ137136ζ139134ζ131213ζ1311132ζ138135ζ138135ζ1311132ζ131213ζ139134ζ137136ζ1310133ζ137136ζ139134    orthogonal lifted from D13
ρ1122200ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ1310133ζ137136ζ1311132ζ139134ζ139134ζ138135ζ131213ζ1310133ζ137136ζ1311132ζ1311132ζ137136ζ1310133ζ131213ζ138135ζ139134ζ138135ζ131213    orthogonal lifted from D13
ρ1222-200ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ1310133ζ137136ζ1311132ζ1391341391341381351312131310133137136131113213111321371361310133131213138135139134ζ138135ζ131213    orthogonal lifted from D26
ρ1322-200ζ137136ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ138135ζ1310133ζ131213ζ13111321311132139134137136138135131013313121313121313101331381351371361391341311132ζ139134ζ137136    orthogonal lifted from D26
ρ1422-200ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ131213ζ137136ζ131213ζ139134ζ1381351381351310133131113213713613121313913413913413121313713613111321310133138135ζ1310133ζ1311132    orthogonal lifted from D26
ρ1522200ζ137136ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ138135ζ1310133ζ131213ζ1311132ζ1311132ζ139134ζ137136ζ138135ζ1310133ζ131213ζ131213ζ1310133ζ138135ζ137136ζ139134ζ1311132ζ139134ζ137136    orthogonal lifted from D13
ρ1622-200ζ139134ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ131213ζ1311132ζ138135ζ13101331310133137136139134131213131113213813513813513111321312131391341371361310133ζ137136ζ139134    orthogonal lifted from D26
ρ1722200ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ138135ζ139134ζ138135ζ137136ζ131213ζ131213ζ1311132ζ1310133ζ139134ζ138135ζ137136ζ137136ζ138135ζ139134ζ1310133ζ1311132ζ131213ζ1311132ζ1310133    orthogonal lifted from D13
ρ182-2000ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ13913413111321391341310133137136137136ζ131213138135ζ1311132139134ζ13101331310133ζ1391341311132ζ138135131213ζ137136131213138135    complex faithful
ρ192-2000ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ131213137136131213139134138135ζ1381351310133ζ1311132137136131213ζ139134139134ζ131213ζ1371361311132ζ131013313813513101331311132    complex faithful
ρ202-2000ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ13913413111321391341310133137136ζ137136131213ζ1381351311132ζ1391341310133ζ1310133139134ζ1311132138135ζ131213137136131213138135    complex faithful
ρ212-2000ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ13713613101331371361311132139134139134ζ138135ζ1312131310133ζ137136ζ13111321311132137136ζ1310133131213138135ζ139134138135131213    complex faithful
ρ222-2000ζ137136ζ131213ζ1311132ζ139134ζ138135ζ131013313813513101331312131311132ζ1311132ζ139134ζ1371361381351310133131213ζ131213ζ1310133ζ1381351371361391341311132139134137136    complex faithful
ρ232-2000ζ139134ζ138135ζ1310133ζ137136ζ131213ζ131113213121313111321381351310133ζ1310133ζ137136139134131213ζ1311132ζ1381351381351311132ζ131213ζ1391341371361310133137136139134    complex faithful
ρ242-2000ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ131213137136131213139134138135138135ζ13101331311132ζ137136ζ131213139134ζ139134131213137136ζ13111321310133ζ13813513101331311132    complex faithful
ρ252-2000ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ138135139134138135137136131213ζ131213ζ1311132ζ1310133ζ139134ζ138135ζ1371361371361381351391341310133131113213121313111321310133    complex faithful
ρ262-2000ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ13713613101331371361311132139134ζ139134138135131213ζ13101331371361311132ζ1311132ζ1371361310133ζ131213ζ138135139134138135131213    complex faithful
ρ272-2000ζ137136ζ131213ζ1311132ζ139134ζ138135ζ1310133138135131013313121313111321311132139134137136ζ138135ζ1310133ζ1312131312131310133138135ζ137136ζ139134ζ1311132139134137136    complex faithful
ρ282-2000ζ139134ζ138135ζ1310133ζ137136ζ131213ζ1311132131213131113213813513101331310133137136ζ139134ζ1312131311132138135ζ138135ζ1311132131213139134ζ137136ζ1310133137136139134    complex faithful
ρ292-2000ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ13813513913413813513713613121313121313111321310133139134138135137136ζ137136ζ138135ζ139134ζ1310133ζ1311132ζ13121313111321310133    complex faithful

Smallest permutation representation of C13⋊D4
On 52 points
Generators in S52
(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)(40 41 42 43 44 45 46 47 48 49 50 51 52)
(1 27 18 43)(2 39 19 42)(3 38 20 41)(4 37 21 40)(5 36 22 52)(6 35 23 51)(7 34 24 50)(8 33 25 49)(9 32 26 48)(10 31 14 47)(11 30 15 46)(12 29 16 45)(13 28 17 44)
(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 43)(28 42)(29 41)(30 40)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)

G:=sub<Sym(52)| (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)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,27,18,43)(2,39,19,42)(3,38,20,41)(4,37,21,40)(5,36,22,52)(6,35,23,51)(7,34,24,50)(8,33,25,49)(9,32,26,48)(10,31,14,47)(11,30,15,46)(12,29,16,45)(13,28,17,44), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,43)(28,42)(29,41)(30,40)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)>;

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)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,27,18,43)(2,39,19,42)(3,38,20,41)(4,37,21,40)(5,36,22,52)(6,35,23,51)(7,34,24,50)(8,33,25,49)(9,32,26,48)(10,31,14,47)(11,30,15,46)(12,29,16,45)(13,28,17,44), (2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,43)(28,42)(29,41)(30,40)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44) );

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),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52)], [(1,27,18,43),(2,39,19,42),(3,38,20,41),(4,37,21,40),(5,36,22,52),(6,35,23,51),(7,34,24,50),(8,33,25,49),(9,32,26,48),(10,31,14,47),(11,30,15,46),(12,29,16,45),(13,28,17,44)], [(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,43),(28,42),(29,41),(30,40),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44)])

C13⋊D4 is a maximal subgroup of
D525C2  D4×D13  D42D13  D26⋊C6  C39⋊D4  C13⋊D12  C397D4  C13⋊S4
C13⋊D4 is a maximal quotient of
C26.D4  D26⋊C4  D4⋊D13  D4.D13  Q8⋊D13  C13⋊Q16  C23.D13  C39⋊D4  C13⋊D12  C397D4

Matrix representation of C13⋊D4 in GL2(𝔽53) generated by

01
5226
,
3118
2922
,
10
2652
G:=sub<GL(2,GF(53))| [0,52,1,26],[31,29,18,22],[1,26,0,52] >;

C13⋊D4 in GAP, Magma, Sage, TeX

C_{13}\rtimes D_4
% in TeX

G:=Group("C13:D4");
// GroupNames label

G:=SmallGroup(104,8);
// by ID

G=gap.SmallGroup(104,8);
# by ID

G:=PCGroup([4,-2,-2,-2,-13,49,1539]);
// Polycyclic

G:=Group<a,b,c|a^13=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C13⋊D4 in TeX
Character table of C13⋊D4 in TeX

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