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## G = D4.10D28order 448 = 26·7

### 5th non-split extension by D4 of D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D4.10D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D4.10D14 — D4.10D28
 Lower central C7 — C14 — C2×C28 — D4.10D28
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for D4.10D28
G = < a,b,c,d | a28=b2=c4=1, d2=a14, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd-1=a21c-1 >

Subgroups: 892 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, D4.8D4, C56⋊C2, D56, C4.Dic7, D14⋊C4, D4⋊D7, Q8⋊D7, C4×C28, C7×M4(2), C2×Dic14, C2×Dic14, C2×D28, C4○D28, C4○D28, D42D7, Q8×D7, C7×C4○D4, Dic14⋊C4, C4.12D28, C7×C4≀C2, C4.D28, C8⋊D14, D4⋊D14, D4.10D14, D4.10D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.8D4, C2×D28, D4×D7, C22⋊D28, D4.10D28

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

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

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

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

58 conjugacy classes

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

58 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D7 D14 D14 D14 D28 D28 D4.8D4 D4×D7 D4×D7 D4.10D28 kernel D4.10D28 Dic14⋊C4 C4.12D28 C7×C4≀C2 C4.D28 C8⋊D14 D4⋊D14 D4.10D14 Dic14 D28 C2×Dic7 C7×D4 C7×Q8 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C7 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 3 3 3 3 6 6 2 3 3 12

Matrix representation of D4.10D28 in GL4(𝔽113) generated by

 60 0 0 0 0 60 0 0 0 0 81 0 0 0 0 81
,
 0 0 81 0 0 0 0 81 60 0 0 0 0 60 0 0
,
 1 0 0 0 0 98 0 0 0 0 98 0 0 0 0 112
,
 0 112 0 0 1 0 0 0 0 0 0 112 0 0 1 0
`G:=sub<GL(4,GF(113))| [60,0,0,0,0,60,0,0,0,0,81,0,0,0,0,81],[0,0,60,0,0,0,0,60,81,0,0,0,0,81,0,0],[1,0,0,0,0,98,0,0,0,0,98,0,0,0,0,112],[0,1,0,0,112,0,0,0,0,0,0,1,0,0,112,0] >;`

D4.10D28 in GAP, Magma, Sage, TeX

`D_4._{10}D_{28}`
`% in TeX`

`G:=Group("D4.10D28");`
`// GroupNames label`

`G:=SmallGroup(448,361);`
`// by ID`

`G=gap.SmallGroup(448,361);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,226,1123,136,851,438,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=a^21*c^-1>;`
`// generators/relations`

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