Copied to
clipboard

## G = C8.17Q16order 128 = 27

### 5th non-split extension by C8 of Q16 acting via Q16/Q8=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8.17Q16
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C8⋊4Q8 — C8.17Q16
 Lower central C1 — C22 — C2×C4 — C8.17Q16
 Upper central C1 — C2×C4 — C4×C8 — C8.17Q16
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.17Q16

Generators and relations for C8.17Q16
G = < a,b,c | a8=1, b8=a4, c2=a2b4, bab-1=cac-1=a5, cbc-1=ab7 >

Smallest permutation representation of C8.17Q16
Regular action on 128 points
Generators in S128
```(1 110 64 70 9 102 56 78)(2 103 49 79 10 111 57 71)(3 112 50 72 11 104 58 80)(4 105 51 65 12 97 59 73)(5 98 52 74 13 106 60 66)(6 107 53 67 14 99 61 75)(7 100 54 76 15 108 62 68)(8 109 55 69 16 101 63 77)(17 37 91 123 25 45 83 115)(18 46 92 116 26 38 84 124)(19 39 93 125 27 47 85 117)(20 48 94 118 28 40 86 126)(21 41 95 127 29 33 87 119)(22 34 96 120 30 42 88 128)(23 43 81 113 31 35 89 121)(24 36 82 122 32 44 90 114)
(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)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)
(1 25 52 87)(2 36 53 126)(3 89 54 27)(4 128 55 38)(5 29 56 91)(6 40 57 114)(7 93 58 31)(8 116 59 42)(9 17 60 95)(10 44 61 118)(11 81 62 19)(12 120 63 46)(13 21 64 83)(14 48 49 122)(15 85 50 23)(16 124 51 34)(18 73 96 109)(20 111 82 75)(22 77 84 97)(24 99 86 79)(26 65 88 101)(28 103 90 67)(30 69 92 105)(32 107 94 71)(33 70 123 106)(35 108 125 72)(37 74 127 110)(39 112 113 76)(41 78 115 98)(43 100 117 80)(45 66 119 102)(47 104 121 68)```

`G:=sub<Sym(128)| (1,110,64,70,9,102,56,78)(2,103,49,79,10,111,57,71)(3,112,50,72,11,104,58,80)(4,105,51,65,12,97,59,73)(5,98,52,74,13,106,60,66)(6,107,53,67,14,99,61,75)(7,100,54,76,15,108,62,68)(8,109,55,69,16,101,63,77)(17,37,91,123,25,45,83,115)(18,46,92,116,26,38,84,124)(19,39,93,125,27,47,85,117)(20,48,94,118,28,40,86,126)(21,41,95,127,29,33,87,119)(22,34,96,120,30,42,88,128)(23,43,81,113,31,35,89,121)(24,36,82,122,32,44,90,114), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128), (1,25,52,87)(2,36,53,126)(3,89,54,27)(4,128,55,38)(5,29,56,91)(6,40,57,114)(7,93,58,31)(8,116,59,42)(9,17,60,95)(10,44,61,118)(11,81,62,19)(12,120,63,46)(13,21,64,83)(14,48,49,122)(15,85,50,23)(16,124,51,34)(18,73,96,109)(20,111,82,75)(22,77,84,97)(24,99,86,79)(26,65,88,101)(28,103,90,67)(30,69,92,105)(32,107,94,71)(33,70,123,106)(35,108,125,72)(37,74,127,110)(39,112,113,76)(41,78,115,98)(43,100,117,80)(45,66,119,102)(47,104,121,68)>;`

`G:=Group( (1,110,64,70,9,102,56,78)(2,103,49,79,10,111,57,71)(3,112,50,72,11,104,58,80)(4,105,51,65,12,97,59,73)(5,98,52,74,13,106,60,66)(6,107,53,67,14,99,61,75)(7,100,54,76,15,108,62,68)(8,109,55,69,16,101,63,77)(17,37,91,123,25,45,83,115)(18,46,92,116,26,38,84,124)(19,39,93,125,27,47,85,117)(20,48,94,118,28,40,86,126)(21,41,95,127,29,33,87,119)(22,34,96,120,30,42,88,128)(23,43,81,113,31,35,89,121)(24,36,82,122,32,44,90,114), (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128), (1,25,52,87)(2,36,53,126)(3,89,54,27)(4,128,55,38)(5,29,56,91)(6,40,57,114)(7,93,58,31)(8,116,59,42)(9,17,60,95)(10,44,61,118)(11,81,62,19)(12,120,63,46)(13,21,64,83)(14,48,49,122)(15,85,50,23)(16,124,51,34)(18,73,96,109)(20,111,82,75)(22,77,84,97)(24,99,86,79)(26,65,88,101)(28,103,90,67)(30,69,92,105)(32,107,94,71)(33,70,123,106)(35,108,125,72)(37,74,127,110)(39,112,113,76)(41,78,115,98)(43,100,117,80)(45,66,119,102)(47,104,121,68) );`

`G=PermutationGroup([[(1,110,64,70,9,102,56,78),(2,103,49,79,10,111,57,71),(3,112,50,72,11,104,58,80),(4,105,51,65,12,97,59,73),(5,98,52,74,13,106,60,66),(6,107,53,67,14,99,61,75),(7,100,54,76,15,108,62,68),(8,109,55,69,16,101,63,77),(17,37,91,123,25,45,83,115),(18,46,92,116,26,38,84,124),(19,39,93,125,27,47,85,117),(20,48,94,118,28,40,86,126),(21,41,95,127,29,33,87,119),(22,34,96,120,30,42,88,128),(23,43,81,113,31,35,89,121),(24,36,82,122,32,44,90,114)], [(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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)], [(1,25,52,87),(2,36,53,126),(3,89,54,27),(4,128,55,38),(5,29,56,91),(6,40,57,114),(7,93,58,31),(8,116,59,42),(9,17,60,95),(10,44,61,118),(11,81,62,19),(12,120,63,46),(13,21,64,83),(14,48,49,122),(15,85,50,23),(16,124,51,34),(18,73,96,109),(20,111,82,75),(22,77,84,97),(24,99,86,79),(26,65,88,101),(28,103,90,67),(30,69,92,105),(32,107,94,71),(33,70,123,106),(35,108,125,72),(37,74,127,110),(39,112,113,76),(41,78,115,98),(43,100,117,80),(45,66,119,102),(47,104,121,68)]])`

38 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 8I 8J 16A ··· 16P order 1 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 8 8 16 ··· 16 size 1 1 1 1 1 1 1 1 4 4 8 8 2 ··· 2 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + - image C1 C2 C2 C2 C4 C4 C8 C8 D4 SD16 Q16 M4(2) C4≀C2 D4.C8 C23.C8 kernel C8.17Q16 C16⋊5C4 C4⋊C16 C8⋊4Q8 C4⋊C8 C4×Q8 C4⋊C4 C2×Q8 C2×C8 C8 C8 C2×C4 C4 C2 C2 # reps 1 1 1 1 2 2 4 4 2 2 2 2 4 8 2

Matrix representation of C8.17Q16 in GL4(𝔽17) generated by

 0 1 0 0 13 0 0 0 0 0 4 0 0 0 0 4
,
 13 2 0 0 8 4 0 0 0 0 0 12 0 0 11 12
,
 16 0 0 0 0 1 0 0 0 0 13 8 0 0 0 4
`G:=sub<GL(4,GF(17))| [0,13,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[13,8,0,0,2,4,0,0,0,0,0,11,0,0,12,12],[16,0,0,0,0,1,0,0,0,0,13,0,0,0,8,4] >;`

C8.17Q16 in GAP, Magma, Sage, TeX

`C_8._{17}Q_{16}`
`% in TeX`

`G:=Group("C8.17Q16");`
`// GroupNames label`

`G:=SmallGroup(128,70);`
`// by ID`

`G=gap.SmallGroup(128,70);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,891,436,1018,136,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^8=a^4,c^2=a^2*b^4,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a*b^7>;`
`// generators/relations`

Export

׿
×
𝔽