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## G = M4(2).19D14order 448 = 26·7

### 2nd non-split extension by M4(2) of D14 acting via D14/D7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — M4(2).19D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — C2×D4⋊2D7 — M4(2).19D14
 Lower central C7 — C14 — C2×C14 — M4(2).19D14
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for M4(2).19D14
G = < a,b,c,d | a8=b2=c14=1, d2=a4, bab=a5, cac-1=ab, dad-1=a5b, bc=cb, bd=db, dcd-1=a4c-1 >

Subgroups: 716 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C4.D4, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, M4(2).8C22, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×Dic14, C2×C4×D7, D42D7, C22×Dic7, C2×C7⋊D4, D4×C14, C4.12D28, C28.D4, C7×C4.D4, D7×M4(2), C2×D42D7, M4(2).19D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, C22×D7, M4(2).8C22, C2×C4×D7, D4×D7, D7×C22⋊C4, M4(2).19D14

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

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

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

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

55 conjugacy classes

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

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 D7 D14 D14 C4×D7 M4(2).8C22 D4×D7 M4(2).19D14 kernel M4(2).19D14 C4.12D28 C28.D4 C7×C4.D4 D7×M4(2) C2×D4⋊2D7 C22×Dic7 C2×C7⋊D4 C4×D7 C4.D4 M4(2) C2×D4 C23 C7 C4 C1 # reps 1 2 1 1 2 1 4 4 4 3 6 3 12 2 6 3

Matrix representation of M4(2).19D14 in GL8(𝔽113)

 15 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 0 0 96 112 112 111 0 0 0 0 0 0 98 0 0 0 0 0 0 1 0 0 0 0 0 0 24 8 16 17
,
 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 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 96 0 112 112
,
 0 80 0 80 0 0 0 0 80 0 80 0 0 0 0 0 0 33 0 9 0 0 0 0 33 0 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 96 112 112 111 0 0 0 0 1 0 0 0 0 0 0 0 8 0 105 1
,
 0 33 0 33 0 0 0 0 33 0 33 0 0 0 0 0 0 104 0 80 0 0 0 0 104 0 80 0 0 0 0 0 0 0 0 0 0 0 112 0 0 0 0 0 17 1 1 2 0 0 0 0 1 0 0 0 0 0 0 0 104 112 8 112

`G:=sub<GL(8,GF(113))| [15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,96,0,0,24,0,0,0,0,112,0,1,8,0,0,0,0,112,98,0,16,0,0,0,0,111,0,0,17],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,96,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,112,0,0,0,0,0,0,0,112],[0,80,0,33,0,0,0,0,80,0,33,0,0,0,0,0,0,80,0,9,0,0,0,0,80,0,9,0,0,0,0,0,0,0,0,0,0,96,1,8,0,0,0,0,0,112,0,0,0,0,0,0,1,112,0,105,0,0,0,0,0,111,0,1],[0,33,0,104,0,0,0,0,33,0,104,0,0,0,0,0,0,33,0,80,0,0,0,0,33,0,80,0,0,0,0,0,0,0,0,0,0,17,1,104,0,0,0,0,0,1,0,112,0,0,0,0,112,1,0,8,0,0,0,0,0,2,0,112] >;`

M4(2).19D14 in GAP, Magma, Sage, TeX

`M_4(2)._{19}D_{14}`
`% in TeX`

`G:=Group("M4(2).19D14");`
`// GroupNames label`

`G:=SmallGroup(448,279);`
`// by ID`

`G=gap.SmallGroup(448,279);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,219,58,570,136,438,18822]);`
`// Polycyclic`

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

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