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G = D13.Q16order 416 = 25·13

The non-split extension by D13 of Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic262C4, D26.19D4, D13.2Q16, D13.3SD16, Dic13.3D4, C52.3(C2×C4), C13⋊(Q8⋊C4), Q81(C13⋊C4), (Q8×C13)⋊1C4, D13⋊C8.1C2, (Q8×D13).2C2, C52⋊C4.1C2, C26.7(C22⋊C4), (C4×D13).9C22, C2.8(D13.D4), C4.3(C2×C13⋊C4), SmallGroup(416,84)

Series: Derived Chief Lower central Upper central

C1C52 — D13.Q16
C1C13C26D26C4×D13C52⋊C4 — D13.Q16
C13C26C52 — D13.Q16
C1C2C4Q8

Generators and relations for D13.Q16
 G = < a,b,c,d | a13=b2=c8=1, d2=c4, bab=a-1, cac-1=a5, ad=da, cbc-1=a4b, bd=db, dcd-1=a-1bc-1 >

13C2
13C2
2C4
13C22
13C4
26C4
52C4
13Q8
13C2×C4
26C2×C4
26C2×C4
26C8
26Q8
2C52
2Dic13
4C13⋊C4
13C2×Q8
13C4⋊C4
13C2×C8
2Dic26
2C13⋊C8
2C2×C13⋊C4
2C4×D13
13Q8⋊C4

Character table of D13.Q16

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D13A13B13C26A26B26C52A52B52C52D52E52F52G52H52I
 size 111313242652525226262626444444888888888
ρ111111111111111111111111111111    trivial
ρ21111111-11-1-1-1-1-1111111111111111    linear of order 2
ρ311111-111-11-1-1-1-1111111-1-1111-1-1-1-1    linear of order 2
ρ411111-11-1-1-11111111111-1-1111-1-1-1-1    linear of order 2
ρ511-1-11-1-1-i1ii-i-ii111111-1-1111-1-1-1-1    linear of order 4
ρ611-1-11-1-1i1-i-iii-i111111-1-1111-1-1-1-1    linear of order 4
ρ711-1-111-1i-1-ii-i-ii111111111111111    linear of order 4
ρ811-1-111-1-i-1i-iii-i111111111111111    linear of order 4
ρ92222-20-2000000022222200-2-2-20000    orthogonal lifted from D4
ρ1022-2-2-202000000022222200-2-2-20000    orthogonal lifted from D4
ρ112-2-2200000022-2-2222-2-2-2000000000    symplectic lifted from Q16, Schur index 2
ρ122-2-22000000-2-222222-2-2-2000000000    symplectic lifted from Q16, Schur index 2
ρ132-22-2000000-2--2-2--2222-2-2-2000000000    complex lifted from SD16
ρ142-22-2000000--2-2--2-2222-2-2-2000000000    complex lifted from SD16
ρ1544004400000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ13111310133132ζ131213813513ζ131213813513ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134    orthogonal lifted from C13⋊C4
ρ1644004-400000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ13111310133132ζ131213813513131213813513139137136134ζ13111310133132ζ139137136134ζ1312138135131312138135131311131013313213111310133132139137136134    orthogonal lifted from C2×C13⋊C4
ρ174400-4000000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ139137136134ζ13111310133132ζ131113101331321312138135131391371361341312138135131311131013313213111310133132139137136134ζ139137136134ζ131213813513    orthogonal lifted from D13.D4
ρ184400-4000000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ131213813513ζ139137136134ζ1391371361341311131013313213121381351313111310133132139137136134139137136134ζ131213813513131213813513ζ13111310133132    orthogonal lifted from D13.D4
ρ1944004-400000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ139137136134ζ1311131013313213111310133132131213813513ζ139137136134ζ131213813513ζ1311131013313213111310133132139137136134139137136134131213813513    orthogonal lifted from C2×C13⋊C4
ρ204400-4000000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ131213813513ζ139137136134139137136134ζ1311131013313213121381351313111310133132139137136134ζ139137136134131213813513ζ13121381351313111310133132    orthogonal lifted from D13.D4
ρ214400-4000000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ139137136134ζ1311131013313213111310133132ζ13121381351313913713613413121381351313111310133132ζ13111310133132ζ139137136134139137136134131213813513    orthogonal lifted from D13.D4
ρ2244004400000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ139137136134ζ13111310133132ζ13111310133132ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ139137136134ζ131213813513    orthogonal lifted from C13⋊C4
ρ234400-4000000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ13111310133132ζ13121381351313121381351313913713613413111310133132139137136134131213813513ζ131213813513ζ1311131013313213111310133132ζ139137136134    orthogonal lifted from D13.D4
ρ2444004-400000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ131213813513ζ13913713613413913713613413111310133132ζ131213813513ζ13111310133132ζ13913713613413913713613413121381351313121381351313111310133132    orthogonal lifted from C2×C13⋊C4
ρ2544004400000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ131213813513ζ139137136134ζ139137136134ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ131213813513ζ13111310133132    orthogonal lifted from C13⋊C4
ρ264400-4000000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ13111310133132ζ131213813513ζ131213813513ζ1391371361341311131013313213913713613413121381351313121381351313111310133132ζ13111310133132139137136134    orthogonal lifted from D13.D4
ρ278-80000000000001311+2ζ1310+2ζ133+2ζ132139+2ζ137+2ζ136+2ζ1341312+2ζ138+2ζ135+2ζ13-2ζ139-2ζ137-2ζ136-2ζ134-2ζ1311-2ζ1310-2ζ133-2ζ132-2ζ1312-2ζ138-2ζ135-2ζ13000000000    symplectic faithful, Schur index 2
ρ288-8000000000000139+2ζ137+2ζ136+2ζ1341312+2ζ138+2ζ135+2ζ131311+2ζ1310+2ζ133+2ζ132-2ζ1312-2ζ138-2ζ135-2ζ13-2ζ139-2ζ137-2ζ136-2ζ134-2ζ1311-2ζ1310-2ζ133-2ζ132000000000    symplectic faithful, Schur index 2
ρ298-80000000000001312+2ζ138+2ζ135+2ζ131311+2ζ1310+2ζ133+2ζ132139+2ζ137+2ζ136+2ζ134-2ζ1311-2ζ1310-2ζ133-2ζ132-2ζ1312-2ζ138-2ζ135-2ζ13-2ζ139-2ζ137-2ζ136-2ζ134000000000    symplectic faithful, Schur index 2

Smallest permutation representation of D13.Q16
On 104 points
Generators in S104
(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)(53 54 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(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 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)(53 70)(54 69)(55 68)(56 67)(57 66)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)
(1 73 43 96 23 64 28 82)(2 68 42 101 24 59 27 87)(3 76 41 93 25 54 39 79)(4 71 40 98 26 62 38 84)(5 66 52 103 14 57 37 89)(6 74 51 95 15 65 36 81)(7 69 50 100 16 60 35 86)(8 77 49 92 17 55 34 91)(9 72 48 97 18 63 33 83)(10 67 47 102 19 58 32 88)(11 75 46 94 20 53 31 80)(12 70 45 99 21 61 30 85)(13 78 44 104 22 56 29 90)
(1 73 23 64)(2 74 24 65)(3 75 25 53)(4 76 26 54)(5 77 14 55)(6 78 15 56)(7 66 16 57)(8 67 17 58)(9 68 18 59)(10 69 19 60)(11 70 20 61)(12 71 21 62)(13 72 22 63)(27 95 42 81)(28 96 43 82)(29 97 44 83)(30 98 45 84)(31 99 46 85)(32 100 47 86)(33 101 48 87)(34 102 49 88)(35 103 50 89)(36 104 51 90)(37 92 52 91)(38 93 40 79)(39 94 41 80)

G:=sub<Sym(104)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (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)(27,43)(28,42)(29,41)(30,40)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(53,70)(54,69)(55,68)(56,67)(57,66)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99), (1,73,43,96,23,64,28,82)(2,68,42,101,24,59,27,87)(3,76,41,93,25,54,39,79)(4,71,40,98,26,62,38,84)(5,66,52,103,14,57,37,89)(6,74,51,95,15,65,36,81)(7,69,50,100,16,60,35,86)(8,77,49,92,17,55,34,91)(9,72,48,97,18,63,33,83)(10,67,47,102,19,58,32,88)(11,75,46,94,20,53,31,80)(12,70,45,99,21,61,30,85)(13,78,44,104,22,56,29,90), (1,73,23,64)(2,74,24,65)(3,75,25,53)(4,76,26,54)(5,77,14,55)(6,78,15,56)(7,66,16,57)(8,67,17,58)(9,68,18,59)(10,69,19,60)(11,70,20,61)(12,71,21,62)(13,72,22,63)(27,95,42,81)(28,96,43,82)(29,97,44,83)(30,98,45,84)(31,99,46,85)(32,100,47,86)(33,101,48,87)(34,102,49,88)(35,103,50,89)(36,104,51,90)(37,92,52,91)(38,93,40,79)(39,94,41,80)>;

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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (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)(27,43)(28,42)(29,41)(30,40)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(53,70)(54,69)(55,68)(56,67)(57,66)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99), (1,73,43,96,23,64,28,82)(2,68,42,101,24,59,27,87)(3,76,41,93,25,54,39,79)(4,71,40,98,26,62,38,84)(5,66,52,103,14,57,37,89)(6,74,51,95,15,65,36,81)(7,69,50,100,16,60,35,86)(8,77,49,92,17,55,34,91)(9,72,48,97,18,63,33,83)(10,67,47,102,19,58,32,88)(11,75,46,94,20,53,31,80)(12,70,45,99,21,61,30,85)(13,78,44,104,22,56,29,90), (1,73,23,64)(2,74,24,65)(3,75,25,53)(4,76,26,54)(5,77,14,55)(6,78,15,56)(7,66,16,57)(8,67,17,58)(9,68,18,59)(10,69,19,60)(11,70,20,61)(12,71,21,62)(13,72,22,63)(27,95,42,81)(28,96,43,82)(29,97,44,83)(30,98,45,84)(31,99,46,85)(32,100,47,86)(33,101,48,87)(34,102,49,88)(35,103,50,89)(36,104,51,90)(37,92,52,91)(38,93,40,79)(39,94,41,80) );

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),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(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),(27,43),(28,42),(29,41),(30,40),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(53,70),(54,69),(55,68),(56,67),(57,66),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(79,98),(80,97),(81,96),(82,95),(83,94),(84,93),(85,92),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99)], [(1,73,43,96,23,64,28,82),(2,68,42,101,24,59,27,87),(3,76,41,93,25,54,39,79),(4,71,40,98,26,62,38,84),(5,66,52,103,14,57,37,89),(6,74,51,95,15,65,36,81),(7,69,50,100,16,60,35,86),(8,77,49,92,17,55,34,91),(9,72,48,97,18,63,33,83),(10,67,47,102,19,58,32,88),(11,75,46,94,20,53,31,80),(12,70,45,99,21,61,30,85),(13,78,44,104,22,56,29,90)], [(1,73,23,64),(2,74,24,65),(3,75,25,53),(4,76,26,54),(5,77,14,55),(6,78,15,56),(7,66,16,57),(8,67,17,58),(9,68,18,59),(10,69,19,60),(11,70,20,61),(12,71,21,62),(13,72,22,63),(27,95,42,81),(28,96,43,82),(29,97,44,83),(30,98,45,84),(31,99,46,85),(32,100,47,86),(33,101,48,87),(34,102,49,88),(35,103,50,89),(36,104,51,90),(37,92,52,91),(38,93,40,79),(39,94,41,80)])

Matrix representation of D13.Q16 in GL6(𝔽313)

100000
010000
00312100
00312010
00312001
0021073240102
,
100000
010000
00312000
00239732111
00138145239102
00311722112
,
294460000
2052120000
002253627788
0011162168241
002634619165
0024016225248
,
212110000
2132920000
003021557248
0050267122248
0029815610
00241072309

G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,312,312,312,210,0,0,1,0,0,73,0,0,0,1,0,240,0,0,0,0,1,102],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,312,239,138,311,0,0,0,73,145,72,0,0,0,211,239,211,0,0,0,1,102,2],[294,205,0,0,0,0,46,212,0,0,0,0,0,0,225,11,263,240,0,0,36,162,46,162,0,0,277,168,191,252,0,0,88,241,65,48],[21,213,0,0,0,0,211,292,0,0,0,0,0,0,302,50,298,241,0,0,15,267,15,0,0,0,57,122,61,72,0,0,248,248,0,309] >;

D13.Q16 in GAP, Magma, Sage, TeX

D_{13}.Q_{16}
% in TeX

G:=Group("D13.Q16");
// GroupNames label

G:=SmallGroup(416,84);
// by ID

G=gap.SmallGroup(416,84);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,579,297,69,9221,3473]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations

Export

Subgroup lattice of D13.Q16 in TeX
Character table of D13.Q16 in TeX

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