Copied to
clipboard

## G = M4(2).21D14order 448 = 26·7

### 4th 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).21D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — C2×Q8⋊2D7 — M4(2).21D14
 Lower central C7 — C14 — C2×C14 — M4(2).21D14
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for M4(2).21D14
G = < a,b,c,d | a8=b2=d2=1, c14=a4, bab=a5, cac-1=ab, dad=a5b, bc=cb, bd=db, dcd=a4c13 >

Subgroups: 812 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C4.D4, C4.10D4, C4.10D4, C2×M4(2), C2×C4○D4, C7⋊C8, C56, C4×D7, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, M4(2).8C22, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, Q82D7, Q8×C14, C28.46D4, C28.10D4, C7×C4.10D4, D7×M4(2), C2×Q82D7, M4(2).21D14
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).21D14

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

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

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

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

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 28A ··· 28F 28G ··· 28L 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 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 14 14 28 28 2 2 4 4 7 7 14 2 2 2 4 4 4 4 28 28 28 28 2 2 2 4 4 4 4 ··· 4 8 ··· 8 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).21D14 kernel M4(2).21D14 C28.46D4 C28.10D4 C7×C4.10D4 D7×M4(2) C2×Q8⋊2D7 C2×C4×D7 C2×D28 C4×D7 C4.10D4 M4(2) C2×Q8 C2×C4 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).21D14 in GL8(𝔽113)

 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 112 1 15 83 0 0 0 0 0 0 15 0 0 0 0 0 98 0 0 0 0 0 0 0 98 0 0 1
,
 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 1 0 0 0 0 0 0 0 0 112 0 0 0 0 0 15 98 0 112
,
 25 0 25 0 0 0 0 0 0 88 0 88 0 0 0 0 88 0 79 0 0 0 0 0 0 25 0 34 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 0 112 1 15 83 0 0 0 0 0 0 0 98
,
 88 0 88 0 0 0 0 0 0 25 0 25 0 0 0 0 34 0 25 0 0 0 0 0 0 79 0 88 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 15 0 1 112

`G:=sub<GL(8,GF(113))| [0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,112,0,98,98,0,0,0,0,1,0,0,0,0,0,0,0,15,15,0,0,0,0,0,0,83,0,0,1],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,15,0,0,0,0,0,1,0,98,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[25,0,88,0,0,0,0,0,0,88,0,25,0,0,0,0,25,0,79,0,0,0,0,0,0,88,0,34,0,0,0,0,0,0,0,0,0,15,112,0,0,0,0,0,15,0,1,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,83,98],[88,0,34,0,0,0,0,0,0,25,0,79,0,0,0,0,88,0,25,0,0,0,0,0,0,25,0,88,0,0,0,0,0,0,0,0,1,0,0,15,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,112] >;`

M4(2).21D14 in GAP, Magma, Sage, TeX

`M_4(2)._{21}D_{14}`
`% in TeX`

`G:=Group("M4(2).21D14");`
`// GroupNames label`

`G:=SmallGroup(448,285);`
`// by ID`

`G=gap.SmallGroup(448,285);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,232,219,58,570,136,438,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^14=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d=a^5*b,b*c=c*b,b*d=d*b,d*c*d=a^4*c^13>;`
`// generators/relations`

׿
×
𝔽