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## G = C14.D8order 224 = 25·7

### 2nd non-split extension by C14 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C14.D8
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×D28 — C14.D8
 Lower central C7 — C14 — C28 — C14.D8
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C14.D8
G = < a,b,c | a14=b8=c2=1, bab-1=cac=a-1, cbc=a7b-1 >

Smallest permutation representation of C14.D8
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 63 111 78 18 96 42 44)(2 62 112 77 19 95 29 43)(3 61 99 76 20 94 30 56)(4 60 100 75 21 93 31 55)(5 59 101 74 22 92 32 54)(6 58 102 73 23 91 33 53)(7 57 103 72 24 90 34 52)(8 70 104 71 25 89 35 51)(9 69 105 84 26 88 36 50)(10 68 106 83 27 87 37 49)(11 67 107 82 28 86 38 48)(12 66 108 81 15 85 39 47)(13 65 109 80 16 98 40 46)(14 64 110 79 17 97 41 45)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 112)(42 111)(43 57)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 98)(84 97)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 28A ··· 28R order 1 2 2 2 2 2 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 size 1 1 1 1 28 28 2 2 4 4 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D4 D7 D8 SD16 D14 C4×D7 D28 C7⋊D4 D4⋊D7 Q8⋊D7 kernel C14.D8 C2×C7⋊C8 C7×C4⋊C4 C2×D28 D28 C28 C2×C14 C4⋊C4 C14 C14 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 3 2 2 3 6 6 6 3 3

Matrix representation of C14.D8 in GL4(𝔽113) generated by

 25 25 0 0 88 79 0 0 0 0 112 0 0 0 0 112
,
 104 46 0 0 13 9 0 0 0 0 87 106 0 0 16 0
,
 1 0 0 0 79 112 0 0 0 0 1 0 0 0 77 112
`G:=sub<GL(4,GF(113))| [25,88,0,0,25,79,0,0,0,0,112,0,0,0,0,112],[104,13,0,0,46,9,0,0,0,0,87,16,0,0,106,0],[1,79,0,0,0,112,0,0,0,0,1,77,0,0,0,112] >;`

C14.D8 in GAP, Magma, Sage, TeX

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

`G:=Group("C14.D8");`
`// GroupNames label`

`G:=SmallGroup(224,15);`
`// by ID`

`G=gap.SmallGroup(224,15);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,579,297,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^14=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^7*b^-1>;`
`// generators/relations`

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