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## G = C8×C13⋊C4order 416 = 25·13

### Direct product of C8 and C13⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C8×C13⋊C4
 Chief series C1 — C13 — C26 — D26 — C4×D13 — C4×C13⋊C4 — C8×C13⋊C4
 Lower central C13 — C8×C13⋊C4
 Upper central C1 — C8

Generators and relations for C8×C13⋊C4
G = < a,b,c | a8=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C8×C13⋊C4
On 104 points
Generators in S104
(1 92 40 66 14 79 27 53)(2 93 41 67 15 80 28 54)(3 94 42 68 16 81 29 55)(4 95 43 69 17 82 30 56)(5 96 44 70 18 83 31 57)(6 97 45 71 19 84 32 58)(7 98 46 72 20 85 33 59)(8 99 47 73 21 86 34 60)(9 100 48 74 22 87 35 61)(10 101 49 75 23 88 36 62)(11 102 50 76 24 89 37 63)(12 103 51 77 25 90 38 64)(13 104 52 78 26 91 39 65)
(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 14)(2 22 13 19)(3 17 12 24)(4 25 11 16)(5 20 10 21)(6 15 9 26)(7 23 8 18)(27 40)(28 48 39 45)(29 43 38 50)(30 51 37 42)(31 46 36 47)(32 41 35 52)(33 49 34 44)(53 66)(54 74 65 71)(55 69 64 76)(56 77 63 68)(57 72 62 73)(58 67 61 78)(59 75 60 70)(79 92)(80 100 91 97)(81 95 90 102)(82 103 89 94)(83 98 88 99)(84 93 87 104)(85 101 86 96)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4L 8A 8B 8C 8D 8E ··· 8P 13A 13B 13C 26A 26B 26C 52A ··· 52F 104A ··· 104L order 1 2 2 2 4 4 4 ··· 4 8 8 8 8 8 ··· 8 13 13 13 26 26 26 52 ··· 52 104 ··· 104 size 1 1 13 13 1 1 13 ··· 13 1 1 1 1 13 ··· 13 4 4 4 4 4 4 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 C13⋊C4 C2×C13⋊C4 C4×C13⋊C4 C8×C13⋊C4 kernel C8×C13⋊C4 C8×D13 D13⋊C8 C4×C13⋊C4 C13⋊2C8 C104 C13⋊C8 C2×C13⋊C4 C13⋊C4 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 3 3 6 12

Matrix representation of C8×C13⋊C4 in GL4(𝔽313) generated by

 5 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 312 312 312 28 1 0 0 213 0 1 0 100 0 0 1 284
,
 101 0 183 214 285 0 201 201 100 312 214 183 211 0 29 311
G:=sub<GL(4,GF(313))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[312,1,0,0,312,0,1,0,312,0,0,1,28,213,100,284],[101,285,100,211,0,0,312,0,183,201,214,29,214,201,183,311] >;

C8×C13⋊C4 in GAP, Magma, Sage, TeX

C_8\times C_{13}\rtimes C_4
% in TeX

G:=Group("C8xC13:C4");
// GroupNames label

G:=SmallGroup(416,66);
// by ID

G=gap.SmallGroup(416,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,55,69,9221,3473]);
// Polycyclic

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

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