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G = C8.SD16order 128 = 27

7th 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.SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C8⋊4Q8 — C8.SD16
 Lower central C1 — C22 — C42 — C8.SD16
 Upper central C1 — C22 — C42 — C8.SD16
 Jennings C1 — C22 — C22 — C42 — C8.SD16

Generators and relations for C8.SD16
G = < a,b,c | a8=b8=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b3 >

Subgroups: 160 in 76 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C4×C8, C8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C2×Q16, C4.10D8, C82C8, C84Q8, Q8⋊Q8, C4⋊Q16, C8.SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C8⋊C22, C8.C22, C4⋊SD16, C8.D4, D4.5D4, C8.SD16

Character table of C8.SD16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 2 2 4 8 8 16 16 4 4 4 4 8 8 8 8 8 8 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 -2 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 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 -2 2 -2 -2 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 -2 2 -2 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 complex lifted from C4○D4 ρ14 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 -2 0 2 0 0 0 0 0 0 0 2 -2 0 -√-2 √-2 √-2 0 0 -√-2 complex lifted from SD16 ρ16 2 -2 -2 2 -2 0 2 0 0 0 0 0 0 0 2 -2 0 √-2 -√-2 -√-2 0 0 √-2 complex lifted from SD16 ρ17 2 -2 -2 2 -2 0 2 0 0 0 0 0 0 0 -2 2 0 -√-2 -√-2 √-2 0 0 √-2 complex lifted from SD16 ρ18 2 -2 -2 2 -2 0 2 0 0 0 0 0 0 0 -2 2 0 √-2 √-2 -√-2 0 0 -√-2 complex lifted from SD16 ρ19 4 -4 -4 4 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 4 -4 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ21 4 -4 4 -4 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ22 4 4 -4 -4 0 0 0 0 0 0 0 0 0 2√2 0 0 -2√2 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 -2√2 0 0 2√2 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2

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

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

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

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

Matrix representation of C8.SD16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 10 0 0 0 0 12 0 0 0 0 0 0 0 10 7 0 0 0 0 5 0
,
 8 0 0 0 0 0 0 2 0 0 0 0 0 0 5 11 0 0 0 0 2 12 0 0 0 0 0 0 3 10 0 0 0 0 8 14
,
 0 9 0 0 0 0 2 0 0 0 0 0 0 0 0 0 12 6 0 0 0 0 15 5 0 0 3 10 0 0 0 0 8 14 0 0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,12,0,0,0,0,10,0,0,0,0,0,0,0,10,5,0,0,0,0,7,0],[8,0,0,0,0,0,0,2,0,0,0,0,0,0,5,2,0,0,0,0,11,12,0,0,0,0,0,0,3,8,0,0,0,0,10,14],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,0,0,3,8,0,0,0,0,10,14,0,0,12,15,0,0,0,0,6,5,0,0] >;

C8.SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(128,422);
// by ID

G=gap.SmallGroup(128,422);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,288,422,387,352,1123,136,2804,718,172]);
// Polycyclic

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

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