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## G = D4.8D14order 224 = 25·7

### 3rd non-split extension by D4 of D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4.8D14
 Chief series C1 — C7 — C14 — C28 — D28 — C4○D28 — D4.8D14
 Lower central C7 — C14 — C28 — D4.8D14
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for D4.8D14
G = < a,b,c,d | a4=b2=1, c14=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c13 >

Subgroups: 230 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4○D8, C7⋊C8, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C4○D28, C7×C4○D4, D4.8D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4○D8, C7⋊D4, C22×D7, C2×C7⋊D4, D4.8D14

Smallest permutation representation of D4.8D14
On 112 points
Generators in S112
```(1 71 15 57)(2 72 16 58)(3 73 17 59)(4 74 18 60)(5 75 19 61)(6 76 20 62)(7 77 21 63)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 98 43 112)(30 99 44 85)(31 100 45 86)(32 101 46 87)(33 102 47 88)(34 103 48 89)(35 104 49 90)(36 105 50 91)(37 106 51 92)(38 107 52 93)(39 108 53 94)(40 109 54 95)(41 110 55 96)(42 111 56 97)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)
(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 14 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 87 43 101)(30 100 44 86)(31 85 45 99)(32 98 46 112)(33 111 47 97)(34 96 48 110)(35 109 49 95)(36 94 50 108)(37 107 51 93)(38 92 52 106)(39 105 53 91)(40 90 54 104)(41 103 55 89)(42 88 56 102)(57 84 71 70)(58 69 72 83)(59 82 73 68)(60 67 74 81)(61 80 75 66)(62 65 76 79)(63 78 77 64)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D14 C4○D8 C7⋊D4 C7⋊D4 D4.8D14 kernel D4.8D14 C2×C7⋊C8 D4⋊D7 D4.D7 Q8⋊D7 C7⋊Q16 C4○D28 C7×C4○D4 C28 C2×C14 C4○D4 C2×C4 D4 Q8 C7 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 3 3 3 4 6 6 6

Matrix representation of D4.8D14 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 1 106 0 0 81 112
,
 91 74 0 0 108 22 0 0 0 0 0 9 0 0 88 0
,
 88 54 0 0 33 1 0 0 0 0 98 0 0 0 0 98
,
 88 54 0 0 1 25 0 0 0 0 98 0 0 0 28 15
`G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,81,0,0,106,112],[91,108,0,0,74,22,0,0,0,0,0,88,0,0,9,0],[88,33,0,0,54,1,0,0,0,0,98,0,0,0,0,98],[88,1,0,0,54,25,0,0,0,0,98,28,0,0,0,15] >;`

D4.8D14 in GAP, Magma, Sage, TeX

`D_4._8D_{14}`
`% in TeX`

`G:=Group("D4.8D14");`
`// GroupNames label`

`G:=SmallGroup(224,145);`
`// by ID`

`G=gap.SmallGroup(224,145);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,579,159,69,6917]);`
`// Polycyclic`

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

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