Copied to
clipboard

## G = M5(2)⋊D7order 448 = 26·7

### 3rd semidirect product of M5(2) and D7 acting via D7/C7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — M5(2)⋊D7
 Chief series C1 — C7 — C14 — C28 — C56 — C2×C56 — C2×C8⋊D7 — M5(2)⋊D7
 Lower central C7 — C14 — C2×C14 — M5(2)⋊D7
 Upper central C1 — C4 — C2×C8 — M5(2)

Generators and relations for M5(2)⋊D7
G = < a,b,c,d | a16=b2=c7=d2=1, bab=a9, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 276 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C16, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, D14, C2×C14, M5(2), M5(2), C2×M4(2), C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C23.C8, C7⋊C16, C112, C8⋊D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C28.C8, C7×M5(2), C2×C8⋊D7, M5(2)⋊D7
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), D14, C22⋊C8, C4×D7, D28, C7⋊D4, C23.C8, C8×D7, C8⋊D7, D14⋊C4, D14⋊C8, M5(2)⋊D7

Smallest permutation representation of M5(2)⋊D7
On 112 points
Generators in S112
```(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(98 106)(100 108)(102 110)(104 112)
(1 111 47 74 23 96 58)(2 112 48 75 24 81 59)(3 97 33 76 25 82 60)(4 98 34 77 26 83 61)(5 99 35 78 27 84 62)(6 100 36 79 28 85 63)(7 101 37 80 29 86 64)(8 102 38 65 30 87 49)(9 103 39 66 31 88 50)(10 104 40 67 32 89 51)(11 105 41 68 17 90 52)(12 106 42 69 18 91 53)(13 107 43 70 19 92 54)(14 108 44 71 20 93 55)(15 109 45 72 21 94 56)(16 110 46 73 22 95 57)
(1 58)(2 51)(3 52)(4 61)(5 62)(6 55)(7 56)(8 49)(9 50)(10 59)(11 60)(12 53)(13 54)(14 63)(15 64)(16 57)(17 33)(18 42)(19 43)(20 36)(21 37)(22 46)(23 47)(24 40)(25 41)(26 34)(27 35)(28 44)(29 45)(30 38)(31 39)(32 48)(67 75)(68 76)(71 79)(72 80)(81 104)(82 105)(83 98)(84 99)(85 108)(86 109)(87 102)(88 103)(89 112)(90 97)(91 106)(92 107)(93 100)(94 101)(95 110)(96 111)```

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 8E 8F 14A 14B 14C 14D 14E 14F 16A 16B 16C 16D 16E 16F 16G 16H 28A ··· 28F 28G 28H 28I 56A ··· 56L 56M ··· 56R 112A ··· 112X order 1 2 2 2 4 4 4 4 7 7 7 8 8 8 8 8 8 14 14 14 14 14 14 16 16 16 16 16 16 16 16 28 ··· 28 28 28 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 28 1 1 2 28 2 2 2 2 2 2 2 28 28 2 2 2 4 4 4 4 4 4 4 28 28 28 28 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 D4 D7 M4(2) D14 D28 C7⋊D4 C4×D7 C8⋊D7 C8×D7 C23.C8 M5(2)⋊D7 kernel M5(2)⋊D7 C28.C8 C7×M5(2) C2×C8⋊D7 C2×C7⋊C8 C2×C4×D7 C2×Dic7 C22×D7 C56 M5(2) C28 C2×C8 C8 C8 C2×C4 C4 C22 C7 C1 # reps 1 1 1 1 2 2 4 4 2 3 2 3 6 6 6 12 12 2 12

Matrix representation of M5(2)⋊D7 in GL4(𝔽113) generated by

 98 0 80 71 0 98 42 0 97 107 15 0 98 21 0 15
,
 1 0 0 0 0 1 0 0 0 33 112 0 80 79 0 112
,
 0 1 0 0 112 9 0 0 0 0 34 1 0 0 53 88
,
 0 1 0 0 1 0 0 0 0 0 88 112 0 0 59 25
`G:=sub<GL(4,GF(113))| [98,0,97,98,0,98,107,21,80,42,15,0,71,0,0,15],[1,0,0,80,0,1,33,79,0,0,112,0,0,0,0,112],[0,112,0,0,1,9,0,0,0,0,34,53,0,0,1,88],[0,1,0,0,1,0,0,0,0,0,88,59,0,0,112,25] >;`

M5(2)⋊D7 in GAP, Magma, Sage, TeX

`M_5(2)\rtimes D_7`
`% in TeX`

`G:=Group("M5(2):D7");`
`// GroupNames label`

`G:=SmallGroup(448,71);`
`// by ID`

`G=gap.SmallGroup(448,71);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,100,570,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^16=b^2=c^7=d^2=1,b*a*b=a^9,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽