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G = D13⋊A4order 312 = 23·3·13

The semidirect product of D13 and A4 acting via A4/C22=C3

metabelian, soluble, monomial, A-group

Aliases: D13⋊A4, C13⋊A4⋊C2, C13⋊(C2×A4), (C2×C26)⋊2C6, C22⋊(C13⋊C6), (C22×D13)⋊2C3, SmallGroup(312,51)

Series: Derived Chief Lower central Upper central

C1C2×C26 — D13⋊A4
C1C13C2×C26C13⋊A4 — D13⋊A4
C2×C26 — D13⋊A4
C1

Generators and relations for D13⋊A4
 G = < a,b,c,d,e | a13=b2=c2=d2=e3=1, bab=a-1, ac=ca, ad=da, eae-1=a9, bc=cb, bd=db, ebe-1=a8b, ece-1=cd=dc, ede-1=c >

3C2
13C2
39C2
52C3
39C22
39C22
52C6
3C26
3D13
4C13⋊C3
13C23
13A4
3D26
3D26
4C13⋊C6
13C2×A4

Character table of D13⋊A4

 class 12A2B2C3A3B6A6B13A13B26A26B26C26D26E26F
 size 1313395252525266666666
ρ11111111111111111    trivial
ρ211-1-111-1-111111111    linear of order 2
ρ31111ζ32ζ3ζ3ζ3211111111    linear of order 3
ρ411-1-1ζ3ζ32ζ6ζ6511111111    linear of order 6
ρ511-1-1ζ32ζ3ζ65ζ611111111    linear of order 6
ρ61111ζ3ζ32ζ32ζ311111111    linear of order 3
ρ73-1-31000033-1-1-1-1-1-1    orthogonal lifted from C2×A4
ρ83-13-1000033-1-1-1-1-1-1    orthogonal lifted from A4
ρ966000000-1+13/2-1-13/2-1-13/2-1-13/2-1+13/2-1-13/2-1+13/2-1+13/2    orthogonal lifted from C13⋊C6
ρ106-2000000-1+13/2-1-13/213111381371361351321311138137136135132ζ1312131013913413313ζ131113813713613513213121310139134133131312131013913413313    orthogonal faithful
ρ116-2000000-1-13/2-1+13/21312131013913413313ζ1312131013913413313ζ1311138137136135132131213101391341331313111381371361351321311138137136135132    orthogonal faithful
ρ126-2000000-1-13/2-1+13/2131213101391341331313121310139134133131311138137136135132ζ13121310139134133131311138137136135132ζ1311138137136135132    orthogonal faithful
ρ136-2000000-1+13/2-1-13/21311138137136135132ζ131113813713613513213121310139134133131311138137136135132ζ13121310139134133131312131013913413313    orthogonal faithful
ρ146-2000000-1+13/2-1-13/2ζ13111381371361351321311138137136135132131213101391341331313111381371361351321312131013913413313ζ1312131013913413313    orthogonal faithful
ρ156-2000000-1-13/2-1+13/2ζ1312131013913413313131213101391341331313111381371361351321312131013913413313ζ13111381371361351321311138137136135132    orthogonal faithful
ρ1666000000-1-13/2-1+13/2-1+13/2-1+13/2-1-13/2-1+13/2-1-13/2-1-13/2    orthogonal lifted from C13⋊C6

Smallest permutation representation of D13⋊A4
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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 17)(15 16)(18 26)(19 25)(20 24)(21 23)(27 33)(28 32)(29 31)(34 39)(35 38)(36 37)(40 50)(41 49)(42 48)(43 47)(44 46)(51 52)
(1 37)(2 38)(3 39)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 50)(15 51)(16 52)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 14)(13 15)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 40)(39 41)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(14 46 32)(15 49 28)(16 52 37)(17 42 33)(18 45 29)(19 48 38)(20 51 34)(21 41 30)(22 44 39)(23 47 35)(24 50 31)(25 40 27)(26 43 36)

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

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

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

Matrix representation of D13⋊A4 in GL6(𝔽79)

1610000
6201000
1800100
6100010
1700001
627878787878
,
77777814013
161716787815
64636321765
17171778620
777777151714
111646264
,
37077242477
533934113758
27426684524
057744757
772243662472
26563376065
,
160113311
8550312353
683460482978
0697477469
111040597145
714515234276
,
16001620
620061621
181019330
610045450
167878321878
630162610

G:=sub<GL(6,GF(79))| [16,62,18,61,17,62,1,0,0,0,0,78,0,1,0,0,0,78,0,0,1,0,0,78,0,0,0,1,0,78,0,0,0,0,1,78],[77,16,64,17,77,1,77,17,63,17,77,1,78,16,63,17,77,1,14,78,2,78,15,64,0,78,17,62,17,62,13,15,65,0,14,64],[37,53,2,0,77,26,0,39,74,57,22,5,77,34,26,7,43,63,24,11,68,44,66,37,24,37,45,7,24,60,77,58,24,57,72,65],[16,8,68,0,11,71,0,5,34,69,10,45,11,50,60,74,40,15,3,31,48,7,59,23,3,23,29,74,71,42,11,53,78,69,45,76],[16,62,18,61,16,63,0,0,1,0,78,0,0,0,0,0,78,1,16,61,19,45,32,62,2,62,33,45,18,61,0,1,0,0,78,0] >;

D13⋊A4 in GAP, Magma, Sage, TeX

D_{13}\rtimes A_4
% in TeX

G:=Group("D13:A4");
// GroupNames label

G:=SmallGroup(312,51);
// by ID

G=gap.SmallGroup(312,51);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-13,97,188,7204,909]);
// Polycyclic

G:=Group<a,b,c,d,e|a^13=b^2=c^2=d^2=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^9,b*c=c*b,b*d=d*b,e*b*e^-1=a^8*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

Export

Subgroup lattice of D13⋊A4 in TeX
Character table of D13⋊A4 in TeX

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