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## G = D28.29D4order 448 = 26·7

### 12nd non-split extension by D28 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D28.29D4
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C4○D28 — D4⋊6D14 — D28.29D4
 Lower central C7 — C14 — C28 — D28.29D4
 Upper central C1 — C2 — C2×C4 — C2×SD16

Generators and relations for D28.29D4
G = < a,b,c,d | a28=b2=d2=1, c4=a14, bab=a-1, ac=ca, dad=a15, bc=cb, dbd=a14b, dcd=c3 >

Subgroups: 1316 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×SD16, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×C14, D4○SD16, C8×D7, C8⋊D7, C56⋊C2, D56, Dic28, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C7×SD16, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, Q8×D7, Q8×D7, Q82D7, Q82D7, C2×C7⋊D4, D4×C14, Q8×C14, D28.2C4, D567C2, D7×SD16, D56⋊C2, SD16⋊D7, SD163D7, D4.D14, C28.C23, C14×SD16, D46D14, Q8.10D14, D28.29D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○SD16, D4×D7, C23×D7, C2×D4×D7, D28.29D4

Smallest permutation representation of D28.29D4
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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 33)(30 32)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(57 61)(58 60)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)
(1 42 110 70 15 56 96 84)(2 43 111 71 16 29 97 57)(3 44 112 72 17 30 98 58)(4 45 85 73 18 31 99 59)(5 46 86 74 19 32 100 60)(6 47 87 75 20 33 101 61)(7 48 88 76 21 34 102 62)(8 49 89 77 22 35 103 63)(9 50 90 78 23 36 104 64)(10 51 91 79 24 37 105 65)(11 52 92 80 25 38 106 66)(12 53 93 81 26 39 107 67)(13 54 94 82 27 40 108 68)(14 55 95 83 28 41 109 69)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(29 78)(30 65)(31 80)(32 67)(33 82)(34 69)(35 84)(36 71)(37 58)(38 73)(39 60)(40 75)(41 62)(42 77)(43 64)(44 79)(45 66)(46 81)(47 68)(48 83)(49 70)(50 57)(51 72)(52 59)(53 74)(54 61)(55 76)(56 63)(85 92)(86 107)(87 94)(88 109)(89 96)(90 111)(91 98)(93 100)(95 102)(97 104)(99 106)(101 108)(103 110)(105 112)```

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

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

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

64 conjugacy classes

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

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 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 D4○SD16 D4×D7 D4×D7 D28.29D4 kernel D28.29D4 D28.2C4 D56⋊7C2 D7×SD16 D56⋊C2 SD16⋊D7 SD16⋊3D7 D4.D14 C28.C23 C14×SD16 D4⋊6D14 Q8.10D14 Dic14 D28 C7⋊D4 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C7 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 3 3 12 3 3 2 3 3 12

Matrix representation of D28.29D4 in GL4(𝔽113) generated by

 0 0 54 82 0 0 31 91 59 31 0 0 82 22 0 0
,
 88 79 0 0 25 25 0 0 0 0 88 79 0 0 25 25
,
 100 0 13 0 0 100 0 13 100 0 100 0 0 100 0 100
,
 0 0 22 101 0 0 12 91 22 101 0 0 12 91 0 0
`G:=sub<GL(4,GF(113))| [0,0,59,82,0,0,31,22,54,31,0,0,82,91,0,0],[88,25,0,0,79,25,0,0,0,0,88,25,0,0,79,25],[100,0,100,0,0,100,0,100,13,0,100,0,0,13,0,100],[0,0,22,12,0,0,101,91,22,12,0,0,101,91,0,0] >;`

D28.29D4 in GAP, Magma, Sage, TeX

`D_{28}._{29}D_4`
`% in TeX`

`G:=Group("D28.29D4");`
`// GroupNames label`

`G:=SmallGroup(448,1215);`
`// by ID`

`G=gap.SmallGroup(448,1215);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,185,136,438,235,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=d^2=1,c^4=a^14,b*a*b=a^-1,a*c=c*a,d*a*d=a^15,b*c=c*b,d*b*d=a^14*b,d*c*d=c^3>;`
`// generators/relations`

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