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## G = D8.D14order 448 = 26·7

### 1st non-split extension by D8 of D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D8.D14
 Chief series C1 — C7 — C14 — C28 — C56 — D56 — D56⋊7C2 — D8.D14
 Lower central C7 — C14 — C28 — C56 — D8.D14
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×D8

Generators and relations for D8.D14
G = < a,b,c,d | a8=b2=1, c14=d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c13 >

Subgroups: 484 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, M5(2), D16, SD32, C2×D8, C4○D8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×D4, C22×C14, C16⋊C22, C7⋊C16, C56⋊C2, D56, Dic28, C2×C56, C7×D8, C7×D8, C4○D28, D4×C14, C28.C8, C7⋊D16, D8.D7, D567C2, C14×D8, D8.D14
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C7⋊D4, C22×D7, C16⋊C22, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7, D8.D14

Smallest permutation representation of D8.D14
On 112 points
Generators in S112
```(1 45 22 38 15 31 8 52)(2 46 23 39 16 32 9 53)(3 47 24 40 17 33 10 54)(4 48 25 41 18 34 11 55)(5 49 26 42 19 35 12 56)(6 50 27 43 20 36 13 29)(7 51 28 44 21 37 14 30)(57 89 64 96 71 103 78 110)(58 90 65 97 72 104 79 111)(59 91 66 98 73 105 80 112)(60 92 67 99 74 106 81 85)(61 93 68 100 75 107 82 86)(62 94 69 101 76 108 83 87)(63 95 70 102 77 109 84 88)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(29 43)(31 45)(33 47)(35 49)(37 51)(39 53)(41 55)(57 96)(58 111)(59 98)(60 85)(61 100)(62 87)(63 102)(64 89)(65 104)(66 91)(67 106)(68 93)(69 108)(70 95)(71 110)(72 97)(73 112)(74 99)(75 86)(76 101)(77 88)(78 103)(79 90)(80 105)(81 92)(82 107)(83 94)(84 109)
(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 65 15 79)(2 78 16 64)(3 63 17 77)(4 76 18 62)(5 61 19 75)(6 74 20 60)(7 59 21 73)(8 72 22 58)(9 57 23 71)(10 70 24 84)(11 83 25 69)(12 68 26 82)(13 81 27 67)(14 66 28 80)(29 106 43 92)(30 91 44 105)(31 104 45 90)(32 89 46 103)(33 102 47 88)(34 87 48 101)(35 100 49 86)(36 85 50 99)(37 98 51 112)(38 111 52 97)(39 96 53 110)(40 109 54 95)(41 94 55 108)(42 107 56 93)```

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 7A 7B 7C 8A 8B 8C 14A ··· 14I 14J ··· 14U 16A 16B 16C 16D 28A ··· 28F 56A ··· 56L order 1 2 2 2 2 2 4 4 4 7 7 7 8 8 8 14 ··· 14 14 ··· 14 16 16 16 16 28 ··· 28 56 ··· 56 size 1 1 2 8 8 56 2 2 56 2 2 2 2 2 4 2 ··· 2 8 ··· 8 28 28 28 28 4 ··· 4 4 ··· 4

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 C7⋊D4 C7⋊D4 C16⋊C22 D4⋊D7 D4⋊D7 D8.D14 kernel D8.D14 C28.C8 C7⋊D16 D8.D7 D56⋊7C2 C14×D8 C56 C2×C28 C2×D8 C28 C2×C14 C2×C8 D8 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 2 2 1 1 1 1 3 2 2 3 6 6 6 2 3 3 12

Matrix representation of D8.D14 in GL4(𝔽113) generated by

 0 46 0 0 27 51 0 0 86 31 82 82 0 31 31 82
,
 112 86 0 0 0 1 0 0 19 112 82 82 46 112 82 31
,
 64 33 0 0 12 49 0 0 83 64 0 83 18 64 30 0
,
 30 0 0 94 48 0 30 30 83 64 0 83 95 0 0 83
`G:=sub<GL(4,GF(113))| [0,27,86,0,46,51,31,31,0,0,82,31,0,0,82,82],[112,0,19,46,86,1,112,112,0,0,82,82,0,0,82,31],[64,12,83,18,33,49,64,64,0,0,0,30,0,0,83,0],[30,48,83,95,0,0,64,0,0,30,0,0,94,30,83,83] >;`

D8.D14 in GAP, Magma, Sage, TeX

`D_8.D_{14}`
`% in TeX`

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

`G:=SmallGroup(448,681);`
`// by ID`

`G=gap.SmallGroup(448,681);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,387,675,185,192,1684,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^14=d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^13>;`
`// generators/relations`

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