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## G = D14.D4order 224 = 25·7

### 1st non-split extension by D14 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D14.D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14.D4
 Lower central C7 — C2×C14 — D14.D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for D14.D4
G = < a,b,c,d | a14=b2=c4=1, d2=a7, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd-1=a7c-1 >

Subgroups: 326 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C22.D4, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×C4×D7, C2×C7⋊D4, D14.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C22×D7, C4○D28, D4×D7, D42D7, D14.D4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 C4○D28 D4×D7 D4⋊2D7 kernel D14.D4 Dic7⋊C4 C4⋊Dic7 D14⋊C4 C23.D7 C7×C22⋊C4 C2×C4×D7 C2×C7⋊D4 D14 C22⋊C4 C14 C2×C4 C23 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 2 3 4 6 3 12 3 3

Matrix representation of D14.D4 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 1 4 0 0 5 21
,
 1 0 0 0 0 28 0 0 0 0 21 1 0 0 24 8
,
 0 12 0 0 17 0 0 0 0 0 27 24 0 0 1 2
,
 17 0 0 0 0 12 0 0 0 0 17 0 0 0 0 17
`G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,5,0,0,4,21],[1,0,0,0,0,28,0,0,0,0,21,24,0,0,1,8],[0,17,0,0,12,0,0,0,0,0,27,1,0,0,24,2],[17,0,0,0,0,12,0,0,0,0,17,0,0,0,0,17] >;`

D14.D4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(224,78);`
`// by ID`

`G=gap.SmallGroup(224,78);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,55,218,188,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^14=b^2=c^4=1,d^2=a^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=a^7*c^-1>;`
`// generators/relations`

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