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## G = M4(2).D14order 448 = 26·7

### 12nd 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).D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — D4⋊D14 — M4(2).D14
 Lower central C7 — C14 — C2×C28 — M4(2).D14
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for M4(2).D14
G = < a,b,c,d | a8=b2=c14=1, d2=a6, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 556 in 108 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C28, C28, D14, C2×C14, C2×C14, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C7⋊C8, C7⋊C8, C56, D28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×C14, D4.4D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4⋊D7, Q8⋊D7, C7×M4(2), C7×D8, C7×SD16, C2×D28, D4×C14, C7×C4○D4, C28.53D4, C28.46D4, C28.D4, C2×D4⋊D7, Q8.Dic7, D4⋊D14, C7×C8⋊C22, M4(2).D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.4D4, D4×D7, D42D7, C2×C7⋊D4, Dic7⋊D4, M4(2).D14

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

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

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

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

49 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 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 2 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 56 2 2 4 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 D4.4D4 D4×D7 D4⋊2D7 M4(2).D14 kernel M4(2).D14 C28.53D4 C28.46D4 C28.D4 C2×D4⋊D7 Q8.Dic7 D4⋊D14 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).D14 in GL8(𝔽113)

 23 66 44 96 0 0 0 0 39 11 100 84 0 0 0 0 52 23 57 84 0 0 0 0 38 32 17 22 0 0 0 0 0 0 0 0 83 83 51 62 0 0 0 0 15 15 0 51 0 0 0 0 82 0 15 0 0 0 0 0 0 0 98 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 26 26 1 0 0 0 0 0 87 87 0 1
,
 28 5 112 99 0 0 0 0 61 88 102 12 0 0 0 0 9 38 42 72 0 0 0 0 87 79 96 68 0 0 0 0 0 0 0 0 87 87 111 0 0 0 0 0 0 0 1 1 0 0 0 0 55 112 13 100 0 0 0 0 58 2 100 13
,
 20 9 14 1 0 0 0 0 54 35 101 11 0 0 0 0 2 98 41 71 0 0 0 0 102 54 45 17 0 0 0 0 0 0 0 0 26 26 0 111 0 0 0 0 0 0 1 1 0 0 0 0 57 1 100 13 0 0 0 0 55 111 13 100

`G:=sub<GL(8,GF(113))| [23,39,52,38,0,0,0,0,66,11,23,32,0,0,0,0,44,100,57,17,0,0,0,0,96,84,84,22,0,0,0,0,0,0,0,0,83,15,82,0,0,0,0,0,83,15,0,0,0,0,0,0,51,0,15,98,0,0,0,0,62,51,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,26,87,0,0,0,0,0,112,26,87,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,61,9,87,0,0,0,0,5,88,38,79,0,0,0,0,112,102,42,96,0,0,0,0,99,12,72,68,0,0,0,0,0,0,0,0,87,0,55,58,0,0,0,0,87,0,112,2,0,0,0,0,111,1,13,100,0,0,0,0,0,1,100,13],[20,54,2,102,0,0,0,0,9,35,98,54,0,0,0,0,14,101,41,45,0,0,0,0,1,11,71,17,0,0,0,0,0,0,0,0,26,0,57,55,0,0,0,0,26,0,1,111,0,0,0,0,0,1,100,13,0,0,0,0,111,1,13,100] >;`

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

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

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

`G:=SmallGroup(448,733);`
`// by ID`

`G=gap.SmallGroup(448,733);`
`# by ID`

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

`G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^6,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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