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G = C8⋊D13order 208 = 24·13

3rd semidirect product of C8 and D13 acting via D13/C13=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D13, C1044C2, D26.1C4, C4.13D26, C133M4(2), C52.13C22, Dic13.1C4, C132C84C2, C26.9(C2×C4), C2.3(C4×D13), (C4×D13).2C2, SmallGroup(208,5)

Series: Derived Chief Lower central Upper central

C1C26 — C8⋊D13
C1C13C26C52C4×D13 — C8⋊D13
C13C26 — C8⋊D13
C1C4C8

Generators and relations for C8⋊D13
 G = < a,b,c | a8=b13=c2=1, ab=ba, cac=a5, cbc=b-1 >

26C2
13C22
13C4
2D13
13C2×C4
13C8
13M4(2)

Smallest permutation representation of C8⋊D13
On 104 points
Generators in S104
(1 95 47 68 18 90 38 58)(2 96 48 69 19 91 39 59)(3 97 49 70 20 79 27 60)(4 98 50 71 21 80 28 61)(5 99 51 72 22 81 29 62)(6 100 52 73 23 82 30 63)(7 101 40 74 24 83 31 64)(8 102 41 75 25 84 32 65)(9 103 42 76 26 85 33 53)(10 104 43 77 14 86 34 54)(11 92 44 78 15 87 35 55)(12 93 45 66 16 88 36 56)(13 94 46 67 17 89 37 57)
(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 21)(15 20)(16 19)(17 18)(22 26)(23 25)(27 35)(28 34)(29 33)(30 32)(36 39)(37 38)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(79 92)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)

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

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

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

C8⋊D13 is a maximal subgroup of   D52.3C4  M4(2)×D13  D52.2C4  D8⋊D13  Q8⋊D26  D4.D26  Q16⋊D13
C8⋊D13 is a maximal quotient of   C52.8Q8  C1048C4  D261C8

58 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D13A···13F26A···26F52A···52L104A···104X
order122444888813···1326···2652···52104···104
size112611262226262···22···22···22···2

58 irreducible representations

dim11111122222
type++++++
imageC1C2C2C2C4C4M4(2)D13D26C4×D13C8⋊D13
kernelC8⋊D13C132C8C104C4×D13Dic13D26C13C8C4C2C1
# reps1111222661224

Matrix representation of C8⋊D13 in GL2(𝔽313) generated by

20571
8108
,
521
15033
,
33312
149280
G:=sub<GL(2,GF(313))| [205,8,71,108],[52,150,1,33],[33,149,312,280] >;

C8⋊D13 in GAP, Magma, Sage, TeX

C_8\rtimes D_{13}
% in TeX

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

G:=SmallGroup(208,5);
// by ID

G=gap.SmallGroup(208,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,101,26,42,4804]);
// Polycyclic

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

Export

Subgroup lattice of C8⋊D13 in TeX

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