Copied to
clipboard

G = D13.D4order 208 = 24·13

The non-split extension by D13 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D262C4, D13.2D4, D26.6C22, (C2×C26)⋊1C4, C22⋊(C13⋊C4), C13⋊(C22⋊C4), C26.7(C2×C4), (C22×D13).2C2, (C2×C13⋊C4)⋊C2, C2.7(C2×C13⋊C4), SmallGroup(208,34)

Series: Derived Chief Lower central Upper central

C1C26 — D13.D4
C1C13D13D26C2×C13⋊C4 — D13.D4
C13C26 — D13.D4
C1C2C22

Generators and relations for D13.D4
 G = < a,b,c,d | a13=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a5, cbc-1=dbd-1=a4b, dcd-1=a-1bc-1 >

2C2
13C2
13C2
26C2
13C22
13C22
26C4
26C4
26C22
26C22
2D13
2C26
13C23
13C2×C4
13C2×C4
2C13⋊C4
2D26
2D26
2C13⋊C4
13C22⋊C4

Character table of D13.D4

 class 12A2B2C2D2E4A4B4C4D13A13B13C26A26B26C26D26E26F26G26H26I
 size 11213132626262626444444444444
ρ11111111111111111111111    trivial
ρ211-111-11-11-1111-1-1-1-1111-1-1    linear of order 2
ρ3111111-1-1-1-1111111111111    linear of order 2
ρ411-111-1-11-11111-1-1-1-1111-1-1    linear of order 2
ρ5111-1-1-1-i-iii111111111111    linear of order 4
ρ611-1-1-11-iii-i111-1-1-1-1111-1-1    linear of order 4
ρ7111-1-1-1ii-i-i111111111111    linear of order 4
ρ811-1-1-11i-i-ii111-1-1-1-1111-1-1    linear of order 4
ρ92-20-22000002220000-2-2-200    orthogonal lifted from D4
ρ102-202-2000002220000-2-2-200    orthogonal lifted from D4
ρ1144-40000000ζ131213813513ζ13111310133132ζ13913713613413111310133132139137136134131213813513139137136134ζ131213813513ζ13111310133132ζ13913713613413121381351313111310133132    orthogonal lifted from C2×C13⋊C4
ρ124440000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134    orthogonal lifted from C13⋊C4
ρ134-400000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132139137136134ζ131213813513ζ1391371361341312138135131311131013313213913713613413121381351313111310133132    orthogonal faithful
ρ144440000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513    orthogonal lifted from C13⋊C4
ρ1544-40000000ζ139137136134ζ131213813513ζ131113101331321312138135131311131013313213913713613413111310133132ζ139137136134ζ131213813513ζ13111310133132139137136134131213813513    orthogonal lifted from C2×C13⋊C4
ρ164-400000000ζ131213813513ζ13111310133132ζ13913713613413111310133132ζ13913713613413121381351313913713613413121381351313111310133132139137136134ζ131213813513ζ13111310133132    orthogonal faithful
ρ1744-40000000ζ13111310133132ζ139137136134ζ13121381351313913713613413121381351313111310133132131213813513ζ13111310133132ζ139137136134ζ13121381351313111310133132139137136134    orthogonal lifted from C2×C13⋊C4
ρ184-400000000ζ13111310133132ζ139137136134ζ13121381351313913713613413121381351313111310133132ζ13121381351313111310133132139137136134131213813513ζ13111310133132ζ139137136134    orthogonal faithful
ρ194-400000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ131113101331321312138135131311131013313213913713613413121381351313111310133132139137136134    orthogonal faithful
ρ204-400000000ζ139137136134ζ131213813513ζ13111310133132ζ13121381351313111310133132139137136134ζ1311131013313213913713613413121381351313111310133132ζ139137136134131213813513    orthogonal faithful
ρ214-400000000ζ139137136134ζ131213813513ζ13111310133132131213813513ζ13111310133132ζ1391371361341311131013313213913713613413121381351313111310133132139137136134ζ131213813513    orthogonal faithful
ρ224440000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132    orthogonal lifted from C13⋊C4

Smallest permutation representation of D13.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 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 26)(11 25)(12 24)(13 23)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)
(1 29)(2 37 13 34)(3 32 12 39)(4 27 11 31)(5 35 10 36)(6 30 9 28)(7 38 8 33)(14 44 19 45)(15 52 18 50)(16 47 17 42)(20 40 26 49)(21 48 25 41)(22 43 24 46)(23 51)
(1 51 23 29)(2 46 22 34)(3 41 21 39)(4 49 20 31)(5 44 19 36)(6 52 18 28)(7 47 17 33)(8 42 16 38)(9 50 15 30)(10 45 14 35)(11 40 26 27)(12 48 25 32)(13 43 24 37)

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

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

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

D13.D4 is a maximal subgroup of
D26.D4  D26.4D4  D26.C23  D4×C13⋊C4
D13.D4 is a maximal quotient of
D26.D4  D26⋊C8  Dic13.D4  D26.Q8  D521C4  Dic26⋊C4  D13.Q16  D52⋊C4  D26.4D4  C26.M4(2)  Dic13.4D4

Matrix representation of D13.D4 in GL4(𝔽53) generated by

0100
523800
00228
00256
,
05200
52000
005125
0022
,
0010
0001
1000
385200
,
0010
0001
52000
15100
G:=sub<GL(4,GF(53))| [0,52,0,0,1,38,0,0,0,0,2,25,0,0,28,6],[0,52,0,0,52,0,0,0,0,0,51,2,0,0,25,2],[0,0,1,38,0,0,0,52,1,0,0,0,0,1,0,0],[0,0,52,15,0,0,0,1,1,0,0,0,0,1,0,0] >;

D13.D4 in GAP, Magma, Sage, TeX

D_{13}.D_4
% in TeX

G:=Group("D13.D4");
// GroupNames label

G:=SmallGroup(208,34);
// by ID

G=gap.SmallGroup(208,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,3204,1214]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations

Export

Subgroup lattice of D13.D4 in TeX
Character table of D13.D4 in TeX

׿
×
𝔽