Copied to
clipboard

## G = D8.D7order 224 = 25·7

### The non-split extension by D8 of D7 acting via D7/C7=C2

Aliases: D8.D7, C72SD32, C28.4D4, C14.9D8, C8.5D14, Dic283C2, C56.3C22, C7⋊C162C2, (C7×D8).1C2, C2.5(D4⋊D7), C4.2(C7⋊D4), SmallGroup(224,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D8.D7
 Chief series C1 — C7 — C14 — C28 — C56 — Dic28 — D8.D7
 Lower central C7 — C14 — C28 — C56 — D8.D7
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for D8.D7
G = < a,b,c,d | a8=b2=c7=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Smallest permutation representation of D8.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)
(1 8)(2 7)(3 6)(4 5)(10 16)(11 15)(12 14)(17 23)(18 22)(19 21)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(41 45)(42 44)(46 48)(49 51)(52 56)(53 55)(57 58)(59 64)(60 63)(61 62)(65 72)(66 71)(67 70)(68 69)(73 76)(74 75)(77 80)(78 79)(81 82)(83 88)(84 87)(85 86)(89 94)(90 93)(91 92)(95 96)(97 100)(98 99)(101 104)(102 103)(105 109)(106 108)(110 112)
(1 65 62 103 92 79 82)(2 66 63 104 93 80 83)(3 67 64 97 94 73 84)(4 68 57 98 95 74 85)(5 69 58 99 96 75 86)(6 70 59 100 89 76 87)(7 71 60 101 90 77 88)(8 72 61 102 91 78 81)(9 50 47 107 30 33 20)(10 51 48 108 31 34 21)(11 52 41 109 32 35 22)(12 53 42 110 25 36 23)(13 54 43 111 26 37 24)(14 55 44 112 27 38 17)(15 56 45 105 28 39 18)(16 49 46 106 29 40 19)
(1 44 5 48)(2 43 6 47)(3 42 7 46)(4 41 8 45)(9 63 13 59)(10 62 14 58)(11 61 15 57)(12 60 16 64)(17 99 21 103)(18 98 22 102)(19 97 23 101)(20 104 24 100)(25 77 29 73)(26 76 30 80)(27 75 31 79)(28 74 32 78)(33 93 37 89)(34 92 38 96)(35 91 39 95)(36 90 40 94)(49 67 53 71)(50 66 54 70)(51 65 55 69)(52 72 56 68)(81 105 85 109)(82 112 86 108)(83 111 87 107)(84 110 88 106)```

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

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

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

D8.D7 is a maximal subgroup of   D8⋊D14  D163D7  D7×SD32  SD32⋊D7  D8.D14  C56.30C23  C56.31C23
D8.D7 is a maximal quotient of   C8.5Dic14  C56.6D4  C14.SD32

32 conjugacy classes

 class 1 2A 2B 4A 4B 7A 7B 7C 8A 8B 14A 14B 14C 14D ··· 14I 16A 16B 16C 16D 28A 28B 28C 56A ··· 56F order 1 2 2 4 4 7 7 7 8 8 14 14 14 14 ··· 14 16 16 16 16 28 28 28 56 ··· 56 size 1 1 8 2 56 2 2 2 2 2 2 2 2 8 ··· 8 14 14 14 14 4 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 D4 D7 D8 D14 SD32 C7⋊D4 D4⋊D7 D8.D7 kernel D8.D7 C7⋊C16 Dic28 C7×D8 C28 D8 C14 C8 C7 C4 C2 C1 # reps 1 1 1 1 1 3 2 3 4 6 3 6

Matrix representation of D8.D7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 0 60 0 0 32 51
,
 112 0 0 0 31 1 0 0 0 0 0 60 0 0 81 0
,
 109 0 0 0 44 28 0 0 0 0 1 0 0 0 0 1
,
 76 45 0 0 60 37 0 0 0 0 16 25 0 0 53 97
`G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,0,32,0,0,60,51],[112,31,0,0,0,1,0,0,0,0,0,81,0,0,60,0],[109,44,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[76,60,0,0,45,37,0,0,0,0,16,53,0,0,25,97] >;`

D8.D7 in GAP, Magma, Sage, TeX

`D_8.D_7`
`% in TeX`

`G:=Group("D8.D7");`
`// GroupNames label`

`G:=SmallGroup(224,33);`
`// by ID`

`G=gap.SmallGroup(224,33);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,73,218,116,122,579,297,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^7=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽