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## G = C16.5Q8order 128 = 27

### 4th non-split extension by C16 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C16.5Q8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C4×C16 — C16.5Q8
 Lower central C1 — C2 — C4 — C2×C8 — C16.5Q8
 Upper central C1 — C22 — C42 — C4×C8 — C16.5Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16.5Q8

Generators and relations for C16.5Q8
G = < a,b,c | a16=b4=1, c2=a8b2, ab=ba, cac-1=a7, cbc-1=a8b-1 >

Smallest permutation representation of C16.5Q8
Regular action on 128 points
Generators in S128
```(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 93 28 46)(2 94 29 47)(3 95 30 48)(4 96 31 33)(5 81 32 34)(6 82 17 35)(7 83 18 36)(8 84 19 37)(9 85 20 38)(10 86 21 39)(11 87 22 40)(12 88 23 41)(13 89 24 42)(14 90 25 43)(15 91 26 44)(16 92 27 45)(49 68 126 111)(50 69 127 112)(51 70 128 97)(52 71 113 98)(53 72 114 99)(54 73 115 100)(55 74 116 101)(56 75 117 102)(57 76 118 103)(58 77 119 104)(59 78 120 105)(60 79 121 106)(61 80 122 107)(62 65 123 108)(63 66 124 109)(64 67 125 110)
(1 79 20 98)(2 70 21 105)(3 77 22 112)(4 68 23 103)(5 75 24 110)(6 66 25 101)(7 73 26 108)(8 80 27 99)(9 71 28 106)(10 78 29 97)(11 69 30 104)(12 76 31 111)(13 67 32 102)(14 74 17 109)(15 65 18 100)(16 72 19 107)(33 118 88 49)(34 125 89 56)(35 116 90 63)(36 123 91 54)(37 114 92 61)(38 121 93 52)(39 128 94 59)(40 119 95 50)(41 126 96 57)(42 117 81 64)(43 124 82 55)(44 115 83 62)(45 122 84 53)(46 113 85 60)(47 120 86 51)(48 127 87 58)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + + - + image C1 C2 C2 C2 C2 Q8 D4 D4 Q16 D8 C4○D16 kernel C16.5Q8 C4×C16 C16⋊3C4 C16⋊4C4 C8.5Q8 C16 C42 C2×C8 C8 C2×C4 C2 # reps 1 1 2 2 2 4 1 1 4 4 16

Matrix representation of C16.5Q8 in GL4(𝔽17) generated by

 16 10 0 0 7 16 0 0 0 0 10 1 0 0 16 10
,
 4 0 0 0 0 4 0 0 0 0 0 13 0 0 4 0
,
 11 13 0 0 13 6 0 0 0 0 12 5 0 0 5 5
`G:=sub<GL(4,GF(17))| [16,7,0,0,10,16,0,0,0,0,10,16,0,0,1,10],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,13,0],[11,13,0,0,13,6,0,0,0,0,12,5,0,0,5,5] >;`

C16.5Q8 in GAP, Magma, Sage, TeX

`C_{16}._5Q_8`
`% in TeX`

`G:=Group("C16.5Q8");`
`// GroupNames label`

`G:=SmallGroup(128,985);`
`// by ID`

`G=gap.SmallGroup(128,985);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,512,422,268,1684,242,4037,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^16=b^4=1,c^2=a^8*b^2,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=a^8*b^-1>;`
`// generators/relations`

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