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## G = C28.3C42order 448 = 26·7

### 3rd non-split extension by C28 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C28.3C42
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C22×C28 — C23.21D14 — C28.3C42
 Lower central C7 — C14 — C28 — C28.3C42
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C28.3C42
G = < a,b,c | a28=b4=1, c4=a14, bab-1=a-1, cac-1=a15, cbc-1=a21b >

Subgroups: 452 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2×C42, C42⋊C2, C2×M4(2), C56, C2×Dic7, C2×C28, C22×C14, C426C4, C4×Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C7×M4(2), C22×Dic7, C22×C28, C2×C4×Dic7, C23.21D14, C14×M4(2), C28.3C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C4≀C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C426C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, D284C4, C14.C42, C28.3C42

Smallest permutation representation of C28.3C42
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 81 29 94)(2 80 30 93)(3 79 31 92)(4 78 32 91)(5 77 33 90)(6 76 34 89)(7 75 35 88)(8 74 36 87)(9 73 37 86)(10 72 38 85)(11 71 39 112)(12 70 40 111)(13 69 41 110)(14 68 42 109)(15 67 43 108)(16 66 44 107)(17 65 45 106)(18 64 46 105)(19 63 47 104)(20 62 48 103)(21 61 49 102)(22 60 50 101)(23 59 51 100)(24 58 52 99)(25 57 53 98)(26 84 54 97)(27 83 55 96)(28 82 56 95)
(1 94 36 74 15 108 50 60)(2 109 37 61 16 95 51 75)(3 96 38 76 17 110 52 62)(4 111 39 63 18 97 53 77)(5 98 40 78 19 112 54 64)(6 85 41 65 20 99 55 79)(7 100 42 80 21 86 56 66)(8 87 43 67 22 101 29 81)(9 102 44 82 23 88 30 68)(10 89 45 69 24 103 31 83)(11 104 46 84 25 90 32 70)(12 91 47 71 26 105 33 57)(13 106 48 58 27 92 34 72)(14 93 49 73 28 107 35 59)```

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

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

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

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 1 1 1 1 2 2 14 ··· 14 28 28 28 28 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + - + + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 D7 Dic7 D14 C4≀C2 Dic14 C4×D7 C7⋊D4 D28 D28⋊4C4 kernel C28.3C42 C2×C4×Dic7 C23.21D14 C14×M4(2) C4×Dic7 C4⋊Dic7 C7×M4(2) C2×C28 C2×C28 C22×C14 C2×M4(2) M4(2) C22×C4 C14 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 2 1 1 3 6 3 8 6 12 12 6 12

Matrix representation of C28.3C42 in GL5(𝔽113)

 1 0 0 0 0 0 98 0 0 0 0 18 15 0 0 0 0 0 112 79 0 0 0 34 25
,
 98 0 0 0 0 0 98 88 0 0 0 0 15 0 0 0 0 0 112 0 0 0 0 34 1
,
 1 0 0 0 0 0 112 36 0 0 0 100 1 0 0 0 0 0 112 0 0 0 0 0 112

`G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,98,18,0,0,0,0,15,0,0,0,0,0,112,34,0,0,0,79,25],[98,0,0,0,0,0,98,0,0,0,0,88,15,0,0,0,0,0,112,34,0,0,0,0,1],[1,0,0,0,0,0,112,100,0,0,0,36,1,0,0,0,0,0,112,0,0,0,0,0,112] >;`

C28.3C42 in GAP, Magma, Sage, TeX

`C_{28}._3C_4^2`
`% in TeX`

`G:=Group("C28.3C4^2");`
`// GroupNames label`

`G:=SmallGroup(448,112);`
`// by ID`

`G=gap.SmallGroup(448,112);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,136,1684,851,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^4=1,c^4=a^14,b*a*b^-1=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^21*b>;`
`// generators/relations`

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