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G = D28.2C4order 224 = 25·7

The non-split extension by D28 of C4 acting through Inn(D28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D28.2C4
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C4○D28 — D28.2C4
 Lower central C7 — C14 — D28.2C4
 Upper central C1 — C8 — C2×C8

Generators and relations for D28.2C4
G = < a,b,c | a28=b2=1, c4=a14, bab=a-1, ac=ca, bc=cb >

Subgroups: 206 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D7 [×2], C14, C14, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, Dic7 [×2], C28 [×2], D14 [×2], C2×C14, C8○D4, C7⋊C8 [×2], C56 [×2], Dic14, C4×D7 [×2], D28, C7⋊D4 [×2], C2×C28, C8×D7 [×2], C8⋊D7 [×2], C4.Dic7, C2×C56, C4○D28, D28.2C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D7, C22×C4, D14 [×3], C8○D4, C4×D7 [×2], C22×D7, C2×C4×D7, D28.2C4

Smallest permutation representation of D28.2C4
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 39)(30 38)(31 37)(32 36)(33 35)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(57 65)(58 64)(59 63)(60 62)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(85 89)(86 88)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)
(1 72 31 98 15 58 45 112)(2 73 32 99 16 59 46 85)(3 74 33 100 17 60 47 86)(4 75 34 101 18 61 48 87)(5 76 35 102 19 62 49 88)(6 77 36 103 20 63 50 89)(7 78 37 104 21 64 51 90)(8 79 38 105 22 65 52 91)(9 80 39 106 23 66 53 92)(10 81 40 107 24 67 54 93)(11 82 41 108 25 68 55 94)(12 83 42 109 26 69 56 95)(13 84 43 110 27 70 29 96)(14 57 44 111 28 71 30 97)```

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D7 D14 D14 C8○D4 C4×D7 C4×D7 D28.2C4 kernel D28.2C4 C8×D7 C8⋊D7 C4.Dic7 C2×C56 C4○D28 Dic14 D28 C7⋊D4 C2×C8 C8 C2×C4 C7 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 4 3 6 3 4 6 6 24

Matrix representation of D28.2C4 in GL4(𝔽113) generated by

 89 79 0 0 10 0 0 0 0 0 98 28 0 0 0 15
,
 0 34 0 0 10 0 0 0 0 0 112 0 0 0 7 1
,
 1 0 0 0 0 1 0 0 0 0 95 0 0 0 0 95
`G:=sub<GL(4,GF(113))| [89,10,0,0,79,0,0,0,0,0,98,0,0,0,28,15],[0,10,0,0,34,0,0,0,0,0,112,7,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,95,0,0,0,0,95] >;`

D28.2C4 in GAP, Magma, Sage, TeX

`D_{28}._2C_4`
`% in TeX`

`G:=Group("D28.2C4");`
`// GroupNames label`

`G:=SmallGroup(224,96);`
`// by ID`

`G=gap.SmallGroup(224,96);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,50,69,6917]);`
`// Polycyclic`

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

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