Copied to
clipboard

## G = D13.D8order 416 = 25·13

### The non-split extension by D13 of D8 acting via D8/C8=C2

Aliases: C1041C4, D13.1D8, D26.9D4, D13.1Q16, Dic13.3Q8, C81(C13⋊C4), C13⋊(C2.D8), C132C86C4, C52.9(C2×C4), C26.2(C4⋊C4), (C8×D13).3C2, C52⋊C4.4C2, C2.5(C52⋊C4), (C4×D13).26C22, C4.9(C2×C13⋊C4), SmallGroup(416,69)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D13.D8
 Chief series C1 — C13 — C26 — D26 — C4×D13 — C52⋊C4 — D13.D8
 Lower central C13 — C26 — C52 — D13.D8
 Upper central C1 — C2 — C4 — C8

Generators and relations for D13.D8
G = < a,b,c,d | a13=b2=c8=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Smallest permutation representation of D13.D8
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 24)(15 23)(16 22)(17 21)(18 20)(25 26)(27 32)(28 31)(29 30)(33 39)(34 38)(35 37)(40 45)(41 44)(42 43)(46 52)(47 51)(48 50)(53 62)(54 61)(55 60)(56 59)(57 58)(63 65)(66 69)(67 68)(70 78)(71 77)(72 76)(73 75)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)(92 98)(93 97)(94 96)(99 104)(100 103)(101 102)
(1 86 30 68 26 102 43 58)(2 87 31 69 14 103 44 59)(3 88 32 70 15 104 45 60)(4 89 33 71 16 92 46 61)(5 90 34 72 17 93 47 62)(6 91 35 73 18 94 48 63)(7 79 36 74 19 95 49 64)(8 80 37 75 20 96 50 65)(9 81 38 76 21 97 51 53)(10 82 39 77 22 98 52 54)(11 83 27 78 23 99 40 55)(12 84 28 66 24 100 41 56)(13 85 29 67 25 101 42 57)
(1 43)(2 51 13 48)(3 46 12 40)(4 41 11 45)(5 49 10 50)(6 44 9 42)(7 52 8 47)(14 38 25 35)(15 33 24 27)(16 28 23 32)(17 36 22 37)(18 31 21 29)(19 39 20 34)(26 30)(53 57 63 59)(54 65 62 64)(55 60 61 56)(66 78 70 71)(67 73 69 76)(72 74 77 75)(79 98 80 93)(81 101 91 103)(82 96 90 95)(83 104 89 100)(84 99 88 92)(85 94 87 97)(86 102)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + - + + - + + image C1 C2 C2 C4 C4 Q8 D4 D8 Q16 C13⋊C4 C2×C13⋊C4 C52⋊C4 D13.D8 kernel D13.D8 C8×D13 C52⋊C4 C13⋊2C8 C104 Dic13 D26 D13 D13 C8 C4 C2 C1 # reps 1 1 2 2 2 1 1 2 2 3 3 6 12

Matrix representation of D13.D8 in GL6(𝔽313)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 31 32 31 31 0 0 282 282 283 282 0 0 242 242 242 243 0 0 312 312 312 312
,
 312 0 0 0 0 0 0 312 0 0 0 0 0 0 312 281 40 311 0 0 0 31 61 30 0 0 0 71 281 71 0 0 0 1 242 2
,
 253 60 0 0 0 0 253 253 0 0 0 0 0 0 64 68 86 60 0 0 227 223 227 0 0 0 26 8 4 253 0 0 18 18 0 22
,
 0 25 0 0 0 0 25 0 0 0 0 0 0 0 283 0 272 32 0 0 243 0 252 252 0 0 282 312 32 272 0 0 29 0 71 311

`G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,282,242,312,0,0,32,282,242,312,0,0,31,283,242,312,0,0,31,282,243,312],[312,0,0,0,0,0,0,312,0,0,0,0,0,0,312,0,0,0,0,0,281,31,71,1,0,0,40,61,281,242,0,0,311,30,71,2],[253,253,0,0,0,0,60,253,0,0,0,0,0,0,64,227,26,18,0,0,68,223,8,18,0,0,86,227,4,0,0,0,60,0,253,22],[0,25,0,0,0,0,25,0,0,0,0,0,0,0,283,243,282,29,0,0,0,0,312,0,0,0,272,252,32,71,0,0,32,252,272,311] >;`

D13.D8 in GAP, Magma, Sage, TeX

`D_{13}.D_8`
`% in TeX`

`G:=Group("D13.D8");`
`// GroupNames label`

`G:=SmallGroup(416,69);`
`// by ID`

`G=gap.SmallGroup(416,69);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,151,579,69,9221,3473]);`
`// Polycyclic`

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

Export

׿
×
𝔽