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## G = M4(2).22D14order 448 = 26·7

### 5th non-split extension by M4(2) of D14 acting via D14/D7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — M4(2).22D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D4.8D14 — M4(2).22D14
 Lower central C7 — C14 — C28 — M4(2).22D14
 Upper central C1 — C4 — C2×C4 — C4≀C2

Generators and relations for M4(2).22D14
G = < a,b,c,d | a8=b2=c14=1, d2=a6b, bab=a5, cac-1=dad-1=a-1b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 412 in 104 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C42, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C8⋊C4, C4≀C2, C4≀C2, C8.C4, C8○D4, C4○D8, C7⋊C8, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8.26D4, C8×D7, C8⋊D7, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C4×C28, C7×M4(2), C4○D28, C7×C4○D4, C42.D7, Dic14⋊C4, C28.53D4, C7×C4≀C2, D28.C4, Q8.Dic7, D4.8D14, M4(2).22D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C22×D7, C8.26D4, C2×C4×D7, D4×D7, D42D7, Dic74D4, M4(2).22D14

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

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 14A 14B 14C 14D 14E 14F 14G 14H 14I 28A ··· 28F 28G ··· 28U 28V 28W 28X 56A ··· 56F order 1 2 2 2 2 4 4 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 8 8 14 14 14 14 14 14 14 14 14 28 ··· 28 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 4 28 1 1 2 4 4 4 28 2 2 2 4 4 14 14 14 14 28 28 28 28 2 2 2 4 4 4 8 8 8 2 ··· 2 4 ··· 4 8 8 8 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D7 C4○D4 D14 D14 D14 C4×D7 C4×D7 C8.26D4 D4×D7 D4⋊2D7 M4(2).22D14 kernel M4(2).22D14 C42.D7 Dic14⋊C4 C28.53D4 C7×C4≀C2 D28.C4 Q8.Dic7 D4.8D14 D4⋊D7 D4.D7 Q8⋊D7 C7⋊Q16 C7⋊C8 C4≀C2 C2×C14 C42 M4(2) C4○D4 D4 Q8 C7 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 3 2 3 3 3 6 6 2 3 3 12

Matrix representation of M4(2).22D14 in GL4(𝔽113) generated by

 0 112 0 0 98 0 0 0 0 0 0 1 0 0 15 0
,
 112 0 0 0 0 1 0 0 0 0 1 0 0 0 0 112
,
 0 16 0 0 16 0 0 0 0 0 0 105 0 0 8 0
,
 0 0 0 106 0 0 106 0 0 99 0 0 14 0 0 0
`G:=sub<GL(4,GF(113))| [0,98,0,0,112,0,0,0,0,0,0,15,0,0,1,0],[112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,112],[0,16,0,0,16,0,0,0,0,0,0,8,0,0,105,0],[0,0,0,14,0,0,99,0,0,106,0,0,106,0,0,0] >;`

M4(2).22D14 in GAP, Magma, Sage, TeX

`M_4(2)._{22}D_{14}`
`% in TeX`

`G:=Group("M4(2).22D14");`
`// GroupNames label`

`G:=SmallGroup(448,357);`
`// by ID`

`G=gap.SmallGroup(448,357);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,219,58,136,1684,851,438,102,18822]);`
`// Polycyclic`

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

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