Copied to
clipboard

## G = M4(2).25D14order 448 = 26·7

### 8th 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).25D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — C2×C8⋊D7 — M4(2).25D14
 Lower central C7 — C14 — C28 — M4(2).25D14
 Upper central C1 — C4 — C2×C4 — C8.C4

Generators and relations for M4(2).25D14
G = < a,b,c,d | a8=b2=1, c14=d2=a6b, bab=a5, cac-1=a-1b, dad-1=a3b, bc=cb, bd=db, dcd-1=c13 >

Subgroups: 412 in 102 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, Dic7, C28, D14, D14, C2×C14, C8.C4, C8.C4, C2×M4(2), C7⋊C8, C7⋊C8, C56, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, M4(2).C4, C8×D7, C8⋊D7, C8⋊D7, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C2×C4×D7, C56.C4, C28.53D4, C7×C8.C4, C2×C8⋊D7, D7×M4(2), M4(2).25D14
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C4×D7, C22×D7, M4(2).C4, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4, M4(2).25D14

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

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - - + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 Q8 Q8 D7 D14 D14 C4×D7 M4(2).C4 D4×D7 Q8×D7 M4(2).25D14 kernel M4(2).25D14 C56.C4 C28.53D4 C7×C8.C4 C2×C8⋊D7 D7×M4(2) C8⋊D7 C4×D7 C2×Dic7 C22×D7 C8.C4 C2×C8 M4(2) C8 C7 C4 C22 C1 # reps 1 1 2 1 1 2 8 2 1 1 3 3 6 12 2 3 3 12

Matrix representation of M4(2).25D14 in GL4(𝔽113) generated by

 0 0 43 13 0 0 100 70 70 100 0 0 13 43 0 0
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 112
,
 30 75 0 0 38 38 0 0 0 0 2 5 0 0 108 108
,
 75 30 0 0 38 38 0 0 0 0 5 2 0 0 108 108
`G:=sub<GL(4,GF(113))| [0,0,70,13,0,0,100,43,43,100,0,0,13,70,0,0],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[30,38,0,0,75,38,0,0,0,0,2,108,0,0,5,108],[75,38,0,0,30,38,0,0,0,0,5,108,0,0,2,108] >;`

M4(2).25D14 in GAP, Magma, Sage, TeX

`M_4(2)._{25}D_{14}`
`% in TeX`

`G:=Group("M4(2).25D14");`
`// GroupNames label`

`G:=SmallGroup(448,427);`
`// by ID`

`G=gap.SmallGroup(448,427);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,219,58,136,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^14=d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a^3*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^13>;`
`// generators/relations`

׿
×
𝔽