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## G = M4(2).15D14order 448 = 26·7

### 15th 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).15D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — D4⋊D14 — M4(2).15D14
 Lower central C7 — C14 — C2×C28 — M4(2).15D14
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for M4(2).15D14
G = < a,b,c,d | a8=b2=c14=1, d2=a2, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >

Subgroups: 524 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, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C7⋊C8, C56, D28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, D4.3D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4⋊D7, Q8⋊D7, C7×M4(2), C7×SD16, C7×Q16, C2×D28, Q8×C14, C7×C4○D4, C28.53D4, C28.46D4, C28.10D4, C2×Q8⋊D7, Q8.Dic7, D4⋊D14, C7×C8.C22, M4(2).15D14
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).15D14

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

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

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

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

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 14H 14I 28A ··· 28F 28G ··· 28O 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 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 4 56 2 2 4 8 2 2 2 8 14 14 28 28 28 56 2 2 2 4 4 4 8 8 8 4 ··· 4 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).15D14 kernel M4(2).15D14 C28.53D4 C28.46D4 C28.10D4 C2×Q8⋊D7 Q8.Dic7 D4⋊D14 C7×C8.C22 C7⋊C8 C7×D4 C7×Q8 C8.C22 C2×C14 M4(2) C2×Q8 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).15D14 in GL6(𝔽113)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 15 0 0 87 0 0 2 1 7 0 0 0 0 0 112 72 0 0 0 91 0 98
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 64 32 112 0 0 0 62 0 0 112
,
 101 16 0 0 0 0 16 101 0 0 0 0 0 0 69 91 72 0 0 0 88 0 0 81 0 0 11 87 44 53 0 0 105 56 85 0
,
 101 16 0 0 0 0 97 12 0 0 0 0 0 0 18 91 72 111 0 0 25 0 0 32 0 0 83 87 44 109 0 0 107 49 85 51

`G:=sub<GL(6,GF(113))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,15,2,0,0,0,0,0,1,0,91,0,0,0,7,112,0,0,0,87,0,72,98],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,64,62,0,0,0,1,32,0,0,0,0,0,112,0,0,0,0,0,0,112],[101,16,0,0,0,0,16,101,0,0,0,0,0,0,69,88,11,105,0,0,91,0,87,56,0,0,72,0,44,85,0,0,0,81,53,0],[101,97,0,0,0,0,16,12,0,0,0,0,0,0,18,25,83,107,0,0,91,0,87,49,0,0,72,0,44,85,0,0,111,32,109,51] >;`

M4(2).15D14 in GAP, Magma, Sage, TeX

`M_4(2)._{15}D_{14}`
`% in TeX`

`G:=Group("M4(2).15D14");`
`// GroupNames label`

`G:=SmallGroup(448,737);`
`// by ID`

`G=gap.SmallGroup(448,737);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,219,184,1123,297,136,1684,851,438,102,18822]);`
`// 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^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

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