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G = D13⋊C8order 208 = 24·13

The semidirect product of D13 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D13⋊C8, C52.3C4, D26.2C4, Dic13.4C22, C13⋊C83C2, C131(C2×C8), C4.3(C13⋊C4), C26.1(C2×C4), (C4×D13).5C2, C2.1(C2×C13⋊C4), SmallGroup(208,28)

Series: Derived Chief Lower central Upper central

C1C13 — D13⋊C8
C1C13C26Dic13C13⋊C8 — D13⋊C8
C13 — D13⋊C8
C1C4

Generators and relations for D13⋊C8
 G = < a,b,c | a13=b2=c8=1, bab=a-1, cac-1=a5, cbc-1=a4b >

13C2
13C2
13C4
13C22
13C8
13C2×C4
13C8
13C2×C8

Character table of D13⋊C8

 class 12A2B2C4A4B4C4D8A8B8C8D8E8F8G8H13A13B13C26A26B26C52A52B52C52D52E52F
 size 1113131113131313131313131313444444444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-1-1-1-1-1-1-1111111111111    linear of order 2
ρ311-1-1-1-1111-1-1-1-1111111111-1-1-1-1-1-1    linear of order 2
ρ411-1-1-1-111-11111-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ511-1-111-1-1i-i-iii-i-ii111111111111    linear of order 4
ρ611-1-111-1-1-iii-i-iii-i111111111111    linear of order 4
ρ71111-1-1-1-1iii-i-i-i-ii111111-1-1-1-1-1-1    linear of order 4
ρ81111-1-1-1-1-i-i-iiiii-i111111-1-1-1-1-1-1    linear of order 4
ρ91-11-1-ii-iiζ85ζ85ζ8ζ87ζ83ζ87ζ83ζ8111-1-1-1-i-iiii-i    linear of order 8
ρ101-11-1i-ii-iζ83ζ83ζ87ζ8ζ85ζ8ζ85ζ87111-1-1-1ii-i-i-ii    linear of order 8
ρ111-1-11-iii-iζ83ζ87ζ83ζ85ζ8ζ8ζ85ζ87111-1-1-1-i-iiii-i    linear of order 8
ρ121-11-1-ii-iiζ8ζ8ζ85ζ83ζ87ζ83ζ87ζ85111-1-1-1-i-iiii-i    linear of order 8
ρ131-1-11i-i-iiζ85ζ8ζ85ζ83ζ87ζ87ζ83ζ8111-1-1-1ii-i-i-ii    linear of order 8
ρ141-11-1i-ii-iζ87ζ87ζ83ζ85ζ8ζ85ζ8ζ83111-1-1-1ii-i-i-ii    linear of order 8
ρ151-1-11-iii-iζ87ζ83ζ87ζ8ζ85ζ85ζ8ζ83111-1-1-1-i-iiii-i    linear of order 8
ρ161-1-11i-i-iiζ8ζ85ζ8ζ87ζ83ζ83ζ87ζ85111-1-1-1ii-i-i-ii    linear of order 8
ρ174400-4-40000000000ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ1391371361341312138135131311131013313213121381351313111310133132139137136134139137136134    orthogonal lifted from C2×C13⋊C4
ρ184400-4-40000000000ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ131113101331321391371361341312138135131391371361341312138135131311131013313213111310133132    orthogonal lifted from C2×C13⋊C4
ρ194400440000000000ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ131213813513    orthogonal lifted from C13⋊C4
ρ204400440000000000ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132    orthogonal lifted from C13⋊C4
ρ214400-4-40000000000ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ1312138135131311131013313213913713613413111310133132139137136134131213813513131213813513    orthogonal lifted from C2×C13⋊C4
ρ224400440000000000ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ139137136134    orthogonal lifted from C13⋊C4
ρ234-4004i-4i0000000000ζ13111310133132ζ139137136134ζ13121381351313121381351313111310133132139137136134ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ131243ζ13843ζ13543ζ13ζ43ζ131143ζ131043ζ13343ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ1394ζ1374ζ1364ζ134    complex faithful, Schur index 2
ρ244-4004i-4i0000000000ζ131213813513ζ13111310133132ζ13913713613413913713613413121381351313111310133132ζ4ζ1394ζ1374ζ1364ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ43ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ43ζ131143ζ131043ζ13343ζ132ζ4ζ13114ζ13104ζ1334ζ132    complex faithful, Schur index 2
ρ254-4004i-4i0000000000ζ139137136134ζ131213813513ζ1311131013313213111310133132139137136134131213813513ζ4ζ13114ζ13104ζ1334ζ132ζ4ζ1394ζ1374ζ1364ζ134ζ43ζ131143ζ131043ζ13343ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ13124ζ1384ζ1354ζ13    complex faithful, Schur index 2
ρ264-400-4i4i0000000000ζ13111310133132ζ139137136134ζ13121381351313121381351313111310133132139137136134ζ43ζ131243ζ13843ζ13543ζ13ζ43ζ131143ζ131043ζ13343ζ132ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ4ζ1394ζ1374ζ1364ζ134ζ43ζ13943ζ13743ζ13643ζ134    complex faithful, Schur index 2
ρ274-400-4i4i0000000000ζ131213813513ζ13111310133132ζ13913713613413913713613413121381351313111310133132ζ43ζ13943ζ13743ζ13643ζ134ζ43ζ131243ζ13843ζ13543ζ13ζ4ζ1394ζ1374ζ1364ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ4ζ13114ζ13104ζ1334ζ132ζ43ζ131143ζ131043ζ13343ζ132    complex faithful, Schur index 2
ρ284-400-4i4i0000000000ζ139137136134ζ131213813513ζ1311131013313213111310133132139137136134131213813513ζ43ζ131143ζ131043ζ13343ζ132ζ43ζ13943ζ13743ζ13643ζ134ζ4ζ13114ζ13104ζ1334ζ132ζ4ζ1394ζ1374ζ1364ζ134ζ4ζ13124ζ1384ζ1354ζ13ζ43ζ131243ζ13843ζ13543ζ13    complex faithful, Schur index 2

Smallest permutation representation of D13⋊C8
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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 63)(54 62)(55 61)(56 60)(57 59)(64 65)(66 68)(69 78)(70 77)(71 76)(72 75)(73 74)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)
(1 92 40 74 25 79 27 65)(2 100 52 66 26 87 39 57)(3 95 51 71 14 82 38 62)(4 103 50 76 15 90 37 54)(5 98 49 68 16 85 36 59)(6 93 48 73 17 80 35 64)(7 101 47 78 18 88 34 56)(8 96 46 70 19 83 33 61)(9 104 45 75 20 91 32 53)(10 99 44 67 21 86 31 58)(11 94 43 72 22 81 30 63)(12 102 42 77 23 89 29 55)(13 97 41 69 24 84 28 60)

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

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

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

D13⋊C8 is a maximal subgroup of   C8×C13⋊C4  C104⋊C4  D521C4  D13.Q16  D13⋊M4(2)  Dic26.C4  D52.C4
D13⋊C8 is a maximal quotient of   D13⋊C16  D26.C8  C4×C13⋊C8  D26⋊C8  Dic13⋊C8

Matrix representation of D13⋊C8 in GL4(𝔽5) generated by

4403
0131
1334
4420
,
1313
1334
0131
4443
,
2430
0014
2030
2000
G:=sub<GL(4,GF(5))| [4,0,1,4,4,1,3,4,0,3,3,2,3,1,4,0],[1,1,0,4,3,3,1,4,1,3,3,4,3,4,1,3],[2,0,2,2,4,0,0,0,3,1,3,0,0,4,0,0] >;

D13⋊C8 in GAP, Magma, Sage, TeX

D_{13}\rtimes C_8
% in TeX

G:=Group("D13:C8");
// GroupNames label

G:=SmallGroup(208,28);
// by ID

G=gap.SmallGroup(208,28);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,46,42,3204,1214]);
// Polycyclic

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

Export

Subgroup lattice of D13⋊C8 in TeX
Character table of D13⋊C8 in TeX

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