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## G = M4(2).13D14order 448 = 26·7

### 13rd non-split extension by M4(2) of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — M4(2).13D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×Dic14 — D4.9D14 — M4(2).13D14
 Lower central C7 — C14 — C2×C28 — M4(2).13D14
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for M4(2).13D14
G = < a,b,c,d | a8=b2=c14=1, d2=a2, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 428 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4.3D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4.D7, C7⋊Q16, C7×M4(2), C7×D8, C7×SD16, C2×Dic14, D4×C14, C7×C4○D4, C28.53D4, C4.12D28, C28.D4, C2×D4.D7, Q8.Dic7, D4.9D14, C7×C8⋊C22, M4(2).13D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.3D4, D4×D7, D42D7, C2×C7⋊D4, Dic7⋊D4, M4(2).13D14

Smallest permutation representation of M4(2).13D14
On 112 points
Generators in S112
```(1 8 66 55 106 99 48 59)(2 60 49 100 107 56 67 9)(3 10 68 43 108 101 50 61)(4 62 51 102 109 44 69 11)(5 12 70 45 110 103 52 63)(6 64 53 104 111 46 57 13)(7 14 58 47 112 105 54 65)(15 73 80 22 88 31 38 95)(16 96 39 32 89 23 81 74)(17 75 82 24 90 33 40 97)(18 98 41 34 91 25 83 76)(19 77 84 26 92 35 42 85)(20 86 29 36 93 27 71 78)(21 79 72 28 94 37 30 87)
(1 106)(3 108)(5 110)(7 112)(9 100)(11 102)(13 104)(15 88)(17 90)(19 92)(21 94)(23 96)(25 98)(27 86)(30 72)(32 74)(34 76)(36 78)(38 80)(40 82)(42 84)(44 62)(46 64)(48 66)(50 68)(52 70)(54 58)(56 60)
(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 66 89 106 81 48 16)(2 88 49 38 107 15 67 80)(3 37 68 87 108 79 50 28)(4 86 51 36 109 27 69 78)(5 35 70 85 110 77 52 26)(6 98 53 34 111 25 57 76)(7 33 58 97 112 75 54 24)(8 96 55 32 99 23 59 74)(9 31 60 95 100 73 56 22)(10 94 43 30 101 21 61 72)(11 29 62 93 102 71 44 20)(12 92 45 42 103 19 63 84)(13 41 64 91 104 83 46 18)(14 90 47 40 105 17 65 82)```

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

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

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

49 conjugacy classes

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

49 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 C4○D4 D14 D14 D14 C7⋊D4 C7⋊D4 D4.3D4 D4×D7 D4⋊2D7 M4(2).13D14 kernel M4(2).13D14 C28.53D4 C4.12D28 C28.D4 C2×D4.D7 Q8.Dic7 D4.9D14 C7×C8⋊C22 C7⋊C8 C7×D4 C7×Q8 C8⋊C22 C2×C14 M4(2) C2×D4 C4○D4 D4 Q8 C7 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 3 2 3 3 3 6 6 2 3 3 3

Matrix representation of M4(2).13D14 in GL8(𝔽113)

 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 77 0 0 0 0 0 0 0 112 0 0 0 0 1 0 0 0 0 0 0 0 44 112 0 0
,
 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 78 0 25 0 0 0 0 0 0 35 0 88 0 0 0 0 63 0 26 0 0 0 0 0 0 50 0 87 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 48 0 54 0 0 0 0 65 0 59 0 0 0 0 0 0 85 0 65 0 0 0 0 28 0 48 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 7 0 0 0 0 0 26 97 0 0 0 0 0 0 7 87 0 0

`G:=sub<GL(8,GF(113))| [0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,1,44,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,77,112,0,0],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[78,0,63,0,0,0,0,0,0,35,0,50,0,0,0,0,25,0,26,0,0,0,0,0,0,88,0,87,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,65,0,28,0,0,0,0,48,0,85,0,0,0,0,0,0,59,0,48,0,0,0,0,54,0,65,0,0,0,0,0,0,0,0,0,0,0,26,7,0,0,0,0,0,0,97,87,0,0,0,0,0,7,0,0,0,0,0,0,16,0,0,0] >;`

M4(2).13D14 in GAP, Magma, Sage, TeX

`M_4(2)._{13}D_{14}`
`% in TeX`

`G:=Group("M4(2).13D14");`
`// GroupNames label`

`G:=SmallGroup(448,734);`
`// by ID`

`G=gap.SmallGroup(448,734);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,219,1123,297,136,1684,851,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^2,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;`
`// generators/relations`

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