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## G = D8⋊11D14order 448 = 26·7

### 5th semidirect product of D8 and D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D8⋊11D14
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C4○D28 — D4⋊8D14 — D8⋊11D14
 Lower central C7 — C14 — C28 — D8⋊11D14
 Upper central C1 — C2 — C2×C4 — C4○D8

Generators and relations for D811D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, dbd=a6b, dcd=c-1 >

Subgroups: 1364 in 258 conjugacy classes, 99 normal (53 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×SD16, C4○D8, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, D28, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, D4○SD16, C8×D7, C8⋊D7, C56⋊C2, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×Dic14, C2×Dic14, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, Q8×D7, Q82D7, C7×C4○D4, D28.2C4, C2×C56⋊C2, D8⋊D7, D7×SD16, SD163D7, Q16⋊D7, D4⋊D14, D4.9D14, C7×C4○D8, D48D14, D4.10D14, D811D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○SD16, D4×D7, C23×D7, C2×D4×D7, D811D14

Smallest permutation representation of D811D14
On 112 points
Generators in S112
```(1 20 33 43 65 100 94 76)(2 21 34 44 66 101 95 77)(3 22 35 45 67 102 96 78)(4 23 36 46 68 103 97 79)(5 24 37 47 69 104 98 80)(6 25 38 48 70 105 85 81)(7 26 39 49 57 106 86 82)(8 27 40 50 58 107 87 83)(9 28 41 51 59 108 88 84)(10 15 42 52 60 109 89 71)(11 16 29 53 61 110 90 72)(12 17 30 54 62 111 91 73)(13 18 31 55 63 112 92 74)(14 19 32 56 64 99 93 75)
(1 76)(2 44)(3 78)(4 46)(5 80)(6 48)(7 82)(8 50)(9 84)(10 52)(11 72)(12 54)(13 74)(14 56)(15 42)(16 90)(17 30)(18 92)(19 32)(20 94)(21 34)(22 96)(23 36)(24 98)(25 38)(26 86)(27 40)(28 88)(29 110)(31 112)(33 100)(35 102)(37 104)(39 106)(41 108)(43 65)(45 67)(47 69)(49 57)(51 59)(53 61)(55 63)(58 83)(60 71)(62 73)(64 75)(66 77)(68 79)(70 81)(85 105)(87 107)(89 109)(91 111)(93 99)(95 101)(97 103)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 42)(13 41)(14 40)(15 111)(16 110)(17 109)(18 108)(19 107)(20 106)(21 105)(22 104)(23 103)(24 102)(25 101)(26 100)(27 99)(28 112)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 98)(68 97)(69 96)(70 95)(71 73)(74 84)(75 83)(76 82)(77 81)(78 80)```

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

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

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

64 conjugacy classes

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

64 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 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 D14 D4○SD16 D4×D7 D4×D7 D8⋊11D14 kernel D8⋊11D14 D28.2C4 C2×C56⋊C2 D8⋊D7 D7×SD16 SD16⋊3D7 Q16⋊D7 D4⋊D14 D4.9D14 C7×C4○D8 D4⋊8D14 D4.10D14 Dic14 D28 C7⋊D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C7 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 3 3 3 6 3 6 2 3 3 12

Matrix representation of D811D14 in GL4(𝔽113) generated by

 76 71 37 42 42 37 71 76 76 71 76 71 42 37 42 37
,
 76 71 37 42 42 37 71 76 37 42 37 42 71 76 71 76
,
 0 0 4 109 0 0 4 81 109 4 0 0 109 32 0 0
,
 0 0 33 33 0 0 104 80 33 33 0 0 104 80 0 0
`G:=sub<GL(4,GF(113))| [76,42,76,42,71,37,71,37,37,71,76,42,42,76,71,37],[76,42,37,71,71,37,42,76,37,71,37,71,42,76,42,76],[0,0,109,109,0,0,4,32,4,4,0,0,109,81,0,0],[0,0,33,104,0,0,33,80,33,104,0,0,33,80,0,0] >;`

D811D14 in GAP, Magma, Sage, TeX

`D_8\rtimes_{11}D_{14}`
`% in TeX`

`G:=Group("D8:11D14");`
`// GroupNames label`

`G:=SmallGroup(448,1223);`
`// by ID`

`G=gap.SmallGroup(448,1223);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,570,185,136,438,235,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

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