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## G = C8.9SD16order 128 = 27

### 9th non-split extension by C8 of SD16 acting via SD16/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8.9SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊C8 — C8.9SD16
 Lower central C1 — C22 — C42 — C8.9SD16
 Upper central C1 — C22 — C42 — C8.9SD16
 Jennings C1 — C22 — C22 — C42 — C8.9SD16

Generators and relations for C8.9SD16
G = < a,b,c | a8=b8=1, c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b3 >

Subgroups: 192 in 88 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C4×C8, C4×C8, Q8⋊C4, C4.Q8, C4⋊Q8, C4⋊Q8, C2×Q16, C8⋊C8, C4.SD16, C4⋊Q16, C83Q8, C83Q8, C8.9SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C41D4, C2×SD16, C8.C22, C85D4, C8.2D4, C8.9SD16

Character table of C8.9SD16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 1 1 2 2 2 2 2 2 16 16 16 16 4 4 4 4 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 2 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 2 0 0 -2 0 0 -2 2 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 2 -2 -2 -2 0 0 0 0 2 -2 0 0 0 -2 0 0 2 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 0 0 2 0 0 2 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 2 2 0 0 -2 0 0 -2 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 -2 0 0 2 0 0 2 -2 orthogonal lifted from D4 ρ14 2 2 2 2 2 -2 2 -2 -2 -2 0 0 0 0 -2 2 0 0 0 2 0 0 -2 0 0 0 orthogonal lifted from D4 ρ15 2 -2 2 -2 0 -2 0 0 2 0 0 0 0 0 √-2 √-2 √-2 -√-2 0 -√-2 -√-2 -2 -√-2 √-2 2 0 complex lifted from SD16 ρ16 2 -2 2 -2 0 2 0 0 -2 0 0 0 0 0 -√-2 √-2 -√-2 √-2 2 -√-2 -√-2 0 √-2 √-2 0 -2 complex lifted from SD16 ρ17 2 -2 2 -2 0 2 0 0 -2 0 0 0 0 0 -√-2 √-2 √-2 -√-2 -2 -√-2 √-2 0 √-2 -√-2 0 2 complex lifted from SD16 ρ18 2 -2 2 -2 0 -2 0 0 2 0 0 0 0 0 -√-2 -√-2 √-2 -√-2 0 √-2 -√-2 2 √-2 √-2 -2 0 complex lifted from SD16 ρ19 2 -2 2 -2 0 2 0 0 -2 0 0 0 0 0 √-2 -√-2 √-2 -√-2 2 √-2 √-2 0 -√-2 -√-2 0 -2 complex lifted from SD16 ρ20 2 -2 2 -2 0 -2 0 0 2 0 0 0 0 0 -√-2 -√-2 -√-2 √-2 0 √-2 √-2 -2 √-2 -√-2 2 0 complex lifted from SD16 ρ21 2 -2 2 -2 0 -2 0 0 2 0 0 0 0 0 √-2 √-2 -√-2 √-2 0 -√-2 √-2 2 -√-2 -√-2 -2 0 complex lifted from SD16 ρ22 2 -2 2 -2 0 2 0 0 -2 0 0 0 0 0 √-2 -√-2 -√-2 √-2 -2 √-2 -√-2 0 -√-2 √-2 0 2 complex lifted from SD16 ρ23 4 -4 -4 4 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ24 4 4 -4 -4 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ25 4 -4 -4 4 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ26 4 4 -4 -4 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8.9SD16
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 59 22 71 78 29 54 98)(2 64 23 68 79 26 55 103)(3 61 24 65 80 31 56 100)(4 58 17 70 73 28 49 97)(5 63 18 67 74 25 50 102)(6 60 19 72 75 30 51 99)(7 57 20 69 76 27 52 104)(8 62 21 66 77 32 53 101)(9 110 84 125 44 91 115 37)(10 107 85 122 45 96 116 34)(11 112 86 127 46 93 117 39)(12 109 87 124 47 90 118 36)(13 106 88 121 48 95 119 33)(14 111 81 126 41 92 120 38)(15 108 82 123 42 89 113 35)(16 105 83 128 43 94 114 40)
(1 127 5 123)(2 126 6 122)(3 125 7 121)(4 124 8 128)(9 104 13 100)(10 103 14 99)(11 102 15 98)(12 101 16 97)(17 109 21 105)(18 108 22 112)(19 107 23 111)(20 106 24 110)(25 82 29 86)(26 81 30 85)(27 88 31 84)(28 87 32 83)(33 80 37 76)(34 79 38 75)(35 78 39 74)(36 77 40 73)(41 72 45 68)(42 71 46 67)(43 70 47 66)(44 69 48 65)(49 90 53 94)(50 89 54 93)(51 96 55 92)(52 95 56 91)(57 119 61 115)(58 118 62 114)(59 117 63 113)(60 116 64 120)

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

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

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

Matrix representation of C8.9SD16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 10 11 7 0 0 10 8 3 13 0 0 7 13 8 3 0 0 13 4 14 16
,
 2 0 0 0 0 0 1 8 0 0 0 0 0 0 0 0 1 0 0 0 16 16 16 16 0 0 16 0 0 0 0 0 2 2 0 1
,
 3 1 0 0 0 0 9 14 0 0 0 0 0 0 7 0 1 0 0 0 1 11 1 1 0 0 1 0 10 0 0 0 15 14 12 6

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,10,7,13,0,0,10,8,13,4,0,0,11,3,8,14,0,0,7,13,3,16],[2,1,0,0,0,0,0,8,0,0,0,0,0,0,0,16,16,2,0,0,0,16,0,2,0,0,1,16,0,0,0,0,0,16,0,1],[3,9,0,0,0,0,1,14,0,0,0,0,0,0,7,1,1,15,0,0,0,11,0,14,0,0,1,1,10,12,0,0,0,1,0,6] >;

C8.9SD16 in GAP, Magma, Sage, TeX

C_8._9{\rm SD}_{16}
% in TeX

G:=Group("C8.9SD16");
// GroupNames label

G:=SmallGroup(128,448);
// by ID

G=gap.SmallGroup(128,448);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,387,100,1123,136,2804,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=1,c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations

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