Copied to
clipboard

## G = C8.27D8order 128 = 27

### 4th non-split extension by C8 of D8 acting via D8/D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C8.27D8
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C4⋊Q16 — C8.27D8
 Lower central C1 — C2 — C2×C4 — C2×C8 — C8.27D8
 Upper central C1 — C22 — C42 — C4×C8 — C8.27D8
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.27D8

Generators and relations for C8.27D8
G = < a,b,c | a8=b8=1, c2=a5, bab-1=a-1, ac=ca, cbc-1=ab-1 >

Character table of C8.27D8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 2 2 4 16 16 2 2 2 2 4 4 8 8 8 8 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 i i -i -i i -i i i -i -i -i i linear of order 4 ρ6 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 i i -i -i -i i -i -i i i i -i linear of order 4 ρ7 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 -i -i i i i -i i i -i -i -i i linear of order 4 ρ8 1 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 -i -i i i -i i -i -i i i i -i linear of order 4 ρ9 2 2 2 2 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 2 2 -2 -2 0 0 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 -2 2 0 0 0 0 2 -2 -2 2 0 0 √2 -√2 -√2 √2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 √2 √2 -√2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ13 2 -2 2 -2 0 -2 2 0 0 0 0 2 -2 -2 2 0 0 -√2 √2 √2 -√2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ14 2 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 -√2 -√2 √2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ15 2 -2 -2 2 -2 0 0 2 0 0 0 -√2 √2 -√2 √2 -√2 √2 0 0 0 0 ζ167-ζ16 ζ167-ζ16 -ζ167+ζ16 ζ165-ζ163 ζ165-ζ163 -ζ167+ζ16 -ζ165+ζ163 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2 ρ16 2 -2 -2 2 -2 0 0 2 0 0 0 -√2 √2 -√2 √2 -√2 √2 0 0 0 0 -ζ167+ζ16 -ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 -ζ165+ζ163 ζ167-ζ16 ζ165-ζ163 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ17 2 -2 -2 2 -2 0 0 2 0 0 0 √2 -√2 √2 -√2 √2 -√2 0 0 0 0 -ζ165+ζ163 -ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 ζ167-ζ16 ζ165-ζ163 -ζ167+ζ16 -ζ167+ζ16 symplectic lifted from Q32, Schur index 2 ρ18 2 -2 -2 2 -2 0 0 2 0 0 0 √2 -√2 √2 -√2 √2 -√2 0 0 0 0 ζ165-ζ163 ζ165-ζ163 -ζ165+ζ163 -ζ167+ζ16 -ζ167+ζ16 -ζ165+ζ163 ζ167-ζ16 ζ167-ζ16 symplectic lifted from Q32, Schur index 2 ρ19 2 -2 -2 2 2 0 0 -2 0 0 0 -√2 √2 -√2 √2 √2 -√2 0 0 0 0 ζ167+ζ16 ζ1615+ζ169 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ165+ζ163 complex lifted from SD32 ρ20 2 -2 2 -2 0 -2 2 0 0 0 0 -2 2 2 -2 0 0 -√-2 √-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ21 2 -2 2 -2 0 -2 2 0 0 0 0 -2 2 2 -2 0 0 √-2 -√-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ22 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ23 2 -2 -2 2 2 0 0 -2 0 0 0 √2 -√2 √2 -√2 -√2 √2 0 0 0 0 ζ165+ζ163 ζ1613+ζ1611 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ1615+ζ169 complex lifted from SD32 ρ24 2 -2 -2 2 2 0 0 -2 0 0 0 √2 -√2 √2 -√2 -√2 √2 0 0 0 0 ζ1613+ζ1611 ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ167+ζ16 complex lifted from SD32 ρ25 2 -2 -2 2 2 0 0 -2 0 0 0 -√2 √2 -√2 √2 √2 -√2 0 0 0 0 ζ1615+ζ169 ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ1613+ζ1611 complex lifted from SD32 ρ26 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ27 4 -4 4 -4 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.17D4, Schur index 2 ρ29 4 4 -4 -4 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.17D4, Schur index 2

Smallest permutation representation of C8.27D8
Regular action on 128 points
Generators in S128
```(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)(33 43 37 47 41 35 45 39)(34 44 38 48 42 36 46 40)(49 59 53 63 57 51 61 55)(50 60 54 64 58 52 62 56)(65 75 69 79 73 67 77 71)(66 76 70 80 74 68 78 72)(81 91 85 95 89 83 93 87)(82 92 86 96 90 84 94 88)(97 107 101 111 105 99 109 103)(98 108 102 112 106 100 110 104)(113 123 117 127 121 115 125 119)(114 124 118 128 122 116 126 120)
(1 73 34 53 30 111 92 114)(2 121 93 102 31 60 35 80)(3 71 36 51 32 109 94 128)(4 119 95 100 17 58 37 78)(5 69 38 49 18 107 96 126)(6 117 81 98 19 56 39 76)(7 67 40 63 20 105 82 124)(8 115 83 112 21 54 41 74)(9 65 42 61 22 103 84 122)(10 113 85 110 23 52 43 72)(11 79 44 59 24 101 86 120)(12 127 87 108 25 50 45 70)(13 77 46 57 26 99 88 118)(14 125 89 106 27 64 47 68)(15 75 48 55 28 97 90 116)(16 123 91 104 29 62 33 66)
(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)```

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

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

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

Matrix representation of C8.27D8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 14 14 0 0 3 14
,
 15 0 0 0 0 9 0 0 0 0 14 3 0 0 3 3
,
 0 9 0 0 15 0 0 0 0 0 1 10 0 0 7 1
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,14,3,0,0,14,14],[15,0,0,0,0,9,0,0,0,0,14,3,0,0,3,3],[0,15,0,0,9,0,0,0,0,0,1,7,0,0,10,1] >;`

C8.27D8 in GAP, Magma, Sage, TeX

`C_8._{27}D_8`
`% in TeX`

`G:=Group("C8.27D8");`
`// GroupNames label`

`G:=SmallGroup(128,94);`
`// by ID`

`G=gap.SmallGroup(128,94);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,416,2804,1411,172,4037,2028,124]);`
`// Polycyclic`

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

Export

׿
×
𝔽