Copied to
clipboard

## G = C28.21C42order 448 = 26·7

### 14th non-split extension by C28 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C28.21C42
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C22×C28 — C2×C4.Dic7 — C28.21C42
 Lower central C7 — C28 — C28.21C42
 Upper central C1 — C4 — C2×M4(2)

Generators and relations for C28.21C42
G = < a,b,c | a28=1, b4=c4=a14, bab-1=a13, ac=ca, cbc-1=a21b >

Subgroups: 260 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×C28, C2×C28, C22×C14, C4.10C42, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C22×C28, C2×C4.Dic7, C14×M4(2), C28.21C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C4.10C42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, C28.21C42

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

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

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

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

82 conjugacy classes

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

82 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + - + + - image C1 C2 C2 C4 C4 D4 Q8 D7 Dic7 D14 C4×D7 D28 C7⋊D4 Dic14 C4.10C42 C28.21C42 kernel C28.21C42 C2×C4.Dic7 C14×M4(2) C2×C7⋊C8 C2×C56 C2×C28 C22×C14 C2×M4(2) C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C7 C1 # reps 1 2 1 8 4 3 1 3 6 3 12 6 12 6 2 12

Matrix representation of C28.21C42 in GL4(𝔽113) generated by

 81 0 0 0 0 81 0 0 25 0 53 0 25 0 0 53
,
 45 0 40 0 0 0 112 1 62 0 68 0 91 15 68 0
,
 1 40 0 0 90 112 0 0 44 68 0 15 0 68 1 0
`G:=sub<GL(4,GF(113))| [81,0,25,25,0,81,0,0,0,0,53,0,0,0,0,53],[45,0,62,91,0,0,0,15,40,112,68,68,0,1,0,0],[1,90,44,0,40,112,68,68,0,0,0,1,0,0,15,0] >;`

C28.21C42 in GAP, Magma, Sage, TeX

`C_{28}._{21}C_4^2`
`% in TeX`

`G:=Group("C28.21C4^2");`
`// GroupNames label`

`G:=SmallGroup(448,117);`
`// by ID`

`G=gap.SmallGroup(448,117);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,184,1123,136,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=1,b^4=c^4=a^14,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^21*b>;`
`// generators/relations`

׿
×
𝔽