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G = C8×D13order 208 = 24·13

Direct product of C8 and D13

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

Aliases: C8×D13, C1043C2, D26.4C4, C4.12D26, C52.12C22, Dic13.4C4, C133(C2×C8), C132C86C2, C26.8(C2×C4), C2.1(C4×D13), (C4×D13).7C2, SmallGroup(208,4)

Series: Derived Chief Lower central Upper central

C1C13 — C8×D13
C1C13C26C52C4×D13 — C8×D13
C13 — C8×D13
C1C8

Generators and relations for C8×D13
 G = < a,b,c | a8=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

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

Smallest permutation representation of C8×D13
On 104 points
Generators in S104
(1 92 52 72 14 80 35 55)(2 93 40 73 15 81 36 56)(3 94 41 74 16 82 37 57)(4 95 42 75 17 83 38 58)(5 96 43 76 18 84 39 59)(6 97 44 77 19 85 27 60)(7 98 45 78 20 86 28 61)(8 99 46 66 21 87 29 62)(9 100 47 67 22 88 30 63)(10 101 48 68 23 89 31 64)(11 102 49 69 24 90 32 65)(12 103 50 70 25 91 33 53)(13 104 51 71 26 79 34 54)
(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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(53 73)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 78)(62 77)(63 76)(64 75)(65 74)(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)| (1,92,52,72,14,80,35,55)(2,93,40,73,15,81,36,56)(3,94,41,74,16,82,37,57)(4,95,42,75,17,83,38,58)(5,96,43,76,18,84,39,59)(6,97,44,77,19,85,27,60)(7,98,45,78,20,86,28,61)(8,99,46,66,21,87,29,62)(9,100,47,67,22,88,30,63)(10,101,48,68,23,89,31,64)(11,102,49,69,24,90,32,65)(12,103,50,70,25,91,33,53)(13,104,51,71,26,79,34,54), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(53,73)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,78)(62,77)(63,76)(64,75)(65,74)(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( (1,92,52,72,14,80,35,55)(2,93,40,73,15,81,36,56)(3,94,41,74,16,82,37,57)(4,95,42,75,17,83,38,58)(5,96,43,76,18,84,39,59)(6,97,44,77,19,85,27,60)(7,98,45,78,20,86,28,61)(8,99,46,66,21,87,29,62)(9,100,47,67,22,88,30,63)(10,101,48,68,23,89,31,64)(11,102,49,69,24,90,32,65)(12,103,50,70,25,91,33,53)(13,104,51,71,26,79,34,54), (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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(53,73)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,78)(62,77)(63,76)(64,75)(65,74)(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([(1,92,52,72,14,80,35,55),(2,93,40,73,15,81,36,56),(3,94,41,74,16,82,37,57),(4,95,42,75,17,83,38,58),(5,96,43,76,18,84,39,59),(6,97,44,77,19,85,27,60),(7,98,45,78,20,86,28,61),(8,99,46,66,21,87,29,62),(9,100,47,67,22,88,30,63),(10,101,48,68,23,89,31,64),(11,102,49,69,24,90,32,65),(12,103,50,70,25,91,33,53),(13,104,51,71,26,79,34,54)], [(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)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15),(13,14),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(53,73),(54,72),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,78),(62,77),(63,76),(64,75),(65,74),(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
C208⋊C2  D13⋊C16  D26.C8  C104⋊C4  D26.8D4  D13.D8  C104.C4  C104.1C4  D52.3C4  D52.2C4  D83D13  D26.6D4  D104⋊C2
C8×D13 is a maximal quotient of
C208⋊C2  C52.8Q8  D261C8

64 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H13A···13F26A···26F52A···52L104A···104X
order122244448888888813···1326···2652···52104···104
size1113131113131111131313132···22···22···22···2

64 irreducible representations

dim11111112222
type++++++
imageC1C2C2C2C4C4C8D13D26C4×D13C8×D13
kernelC8×D13C132C8C104C4×D13Dic13D26D13C8C4C2C1
# reps1111228661224

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

1880
0188
,
142270
312256
,
291181
2522
G:=sub<GL(2,GF(313))| [188,0,0,188],[142,312,270,256],[291,25,181,22] >;

C8×D13 in GAP, Magma, Sage, TeX

C_8\times D_{13}
% in TeX

G:=Group("C8xD13");
// GroupNames label

G:=SmallGroup(208,4);
// by ID

G=gap.SmallGroup(208,4);
# by ID

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

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

Export

Subgroup lattice of C8×D13 in TeX

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