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## G = D8⋊5D14order 448 = 26·7

### 5th semidirect product of D8 and D14 acting via D14/D7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D8⋊5D14
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C4○D28 — D4⋊6D14 — D8⋊5D14
 Lower central C7 — C14 — C28 — D8⋊5D14
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D85D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, cac-1=a5, dad=a3, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 1548 in 268 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×D8, C4○D8, C8⋊C22, C8⋊C22, 2+ 1+4, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, D28, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C22×D7, C22×C14, D4○D8, C8×D7, C8⋊D7, C56⋊C2, D56, C2×C7⋊C8, D4⋊D7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×D8, C7×SD16, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, Q82D7, C2×C7⋊D4, D4×C14, C7×C4○D4, D28.C4, C8⋊D14, D7×D8, D8⋊D7, D56⋊C2, SD163D7, C2×D4⋊D7, D4.8D14, C7×C8⋊C22, D46D14, D48D14, D85D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○D8, D4×D7, C23×D7, C2×D4×D7, D85D14

Smallest permutation representation of D85D14
On 112 points
Generators in S112
```(1 64 85 30 110 50 23 77)(2 51 86 78 111 65 24 31)(3 66 87 32 112 52 25 79)(4 53 88 80 99 67 26 33)(5 68 89 34 100 54 27 81)(6 55 90 82 101 69 28 35)(7 70 91 36 102 56 15 83)(8 43 92 84 103 57 16 37)(9 58 93 38 104 44 17 71)(10 45 94 72 105 59 18 39)(11 60 95 40 106 46 19 73)(12 47 96 74 107 61 20 41)(13 62 97 42 108 48 21 75)(14 49 98 76 109 63 22 29)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 109)(16 110)(17 111)(18 112)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)
(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)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 42)(25 41)(26 40)(27 39)(28 38)(43 102)(44 101)(45 100)(46 99)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 104)(56 103)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)```

`G:=sub<Sym(112)| (1,64,85,30,110,50,23,77)(2,51,86,78,111,65,24,31)(3,66,87,32,112,52,25,79)(4,53,88,80,99,67,26,33)(5,68,89,34,100,54,27,81)(6,55,90,82,101,69,28,35)(7,70,91,36,102,56,15,83)(8,43,92,84,103,57,16,37)(9,58,93,38,104,44,17,71)(10,45,94,72,105,59,18,39)(11,60,95,40,106,46,19,73)(12,47,96,74,107,61,20,41)(13,62,97,42,108,48,21,75)(14,49,98,76,109,63,22,29), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,109)(16,110)(17,111)(18,112)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (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)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,42)(25,41)(26,40)(27,39)(28,38)(43,102)(44,101)(45,100)(46,99)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)>;`

`G:=Group( (1,64,85,30,110,50,23,77)(2,51,86,78,111,65,24,31)(3,66,87,32,112,52,25,79)(4,53,88,80,99,67,26,33)(5,68,89,34,100,54,27,81)(6,55,90,82,101,69,28,35)(7,70,91,36,102,56,15,83)(8,43,92,84,103,57,16,37)(9,58,93,38,104,44,17,71)(10,45,94,72,105,59,18,39)(11,60,95,40,106,46,19,73)(12,47,96,74,107,61,20,41)(13,62,97,42,108,48,21,75)(14,49,98,76,109,63,22,29), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,109)(16,110)(17,111)(18,112)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (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)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,42)(25,41)(26,40)(27,39)(28,38)(43,102)(44,101)(45,100)(46,99)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91) );`

`G=PermutationGroup([[(1,64,85,30,110,50,23,77),(2,51,86,78,111,65,24,31),(3,66,87,32,112,52,25,79),(4,53,88,80,99,67,26,33),(5,68,89,34,100,54,27,81),(6,55,90,82,101,69,28,35),(7,70,91,36,102,56,15,83),(8,43,92,84,103,57,16,37),(9,58,93,38,104,44,17,71),(10,45,94,72,105,59,18,39),(11,60,95,40,106,46,19,73),(12,47,96,74,107,61,20,41),(13,62,97,42,108,48,21,75),(14,49,98,76,109,63,22,29)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,109),(16,110),(17,111),(18,112),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84)], [(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),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,37),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,42),(25,41),(26,40),(27,39),(28,38),(43,102),(44,101),(45,100),(46,99),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,106),(54,105),(55,104),(56,103),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,98),(78,97),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91)]])`

55 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 8E 14A 14B 14C 14D 14E 14F 14G ··· 14O 28A ··· 28F 28G 28H 28I 56A ··· 56F order 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 8 14 14 14 14 14 14 14 ··· 14 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 4 4 4 14 14 28 28 28 2 2 4 14 14 28 2 2 2 4 4 14 14 28 2 2 2 4 4 4 8 ··· 8 4 ··· 4 8 8 8 8 ··· 8

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 D14 D4○D8 D4×D7 D4×D7 D8⋊5D14 kernel D8⋊5D14 D28.C4 C8⋊D14 D7×D8 D8⋊D7 D56⋊C2 SD16⋊3D7 C2×D4⋊D7 D4.8D14 C7×C8⋊C22 D4⋊6D14 D4⋊8D14 Dic14 D28 C7⋊D4 C8⋊C22 M4(2) D8 SD16 C2×D4 C4○D4 C7 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 3 3 6 6 3 3 2 3 3 3

Matrix representation of D85D14 in GL8(𝔽113)

 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 0 68 99 13 8 0 0 0 0 8 38 16 11 0 0 0 0 89 59 69 69 0 0 0 0 54 50 102 51
,
 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 110 64 0 0 0 0 0 86 70 0 111 0 0 0 0 71 97 110 67 0 0 0 0 88 4 40 43
,
 104 33 0 0 0 0 0 0 80 33 0 0 0 0 0 0 0 0 104 33 0 0 0 0 0 0 80 33 0 0 0 0 0 0 0 0 32 39 64 44 0 0 0 0 27 43 0 2 0 0 0 0 81 74 110 51 0 0 0 0 61 75 40 41
,
 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 0 0 0 109 0 0 0 0 0 0 28 0 0 0 0 0 0 0 73 60 62 99 0 0 0 0 72 35 105 51

`G:=sub<GL(8,GF(113))| [0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,68,8,89,54,0,0,0,0,99,38,59,50,0,0,0,0,13,16,69,102,0,0,0,0,8,11,69,51],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,86,71,88,0,0,0,0,110,70,97,4,0,0,0,0,64,0,110,40,0,0,0,0,0,111,67,43],[104,80,0,0,0,0,0,0,33,33,0,0,0,0,0,0,0,0,104,80,0,0,0,0,0,0,33,33,0,0,0,0,0,0,0,0,32,27,81,61,0,0,0,0,39,43,74,75,0,0,0,0,64,0,110,40,0,0,0,0,44,2,51,41],[0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,0,28,73,72,0,0,0,0,109,0,60,35,0,0,0,0,0,0,62,105,0,0,0,0,0,0,99,51] >;`

D85D14 in GAP, Magma, Sage, TeX

`D_8\rtimes_5D_{14}`
`% in TeX`

`G:=Group("D8:5D14");`
`// GroupNames label`

`G:=SmallGroup(448,1227);`
`// by ID`

`G=gap.SmallGroup(448,1227);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,570,185,438,235,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

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