p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.7SD16, C42.241C23, C4⋊C4.63D4, (C2×C8).93D4, C8⋊2C8.8C2, (C2×Q8).57D4, C8⋊4Q8.1C2, Q8⋊Q8.9C2, C4.83(C2×SD16), C4⋊Q8.62C22, C4⋊C8.185C22, (C4×C8).144C22, C4⋊Q16.13C2, C2.7(C8.D4), (C4×Q8).45C22, C4.123(C8⋊C22), C2.11(C4⋊SD16), C4.10D8.12C2, C4.43(C8.C22), C2.13(D4.5D4), C22.202(C4⋊D4), (C2×C4).26(C4○D4), (C2×C4).1276(C2×D4), SmallGroup(128,422)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
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 [×3], C4 [×2], C4 [×2], C4 [×5], C22, C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×6], C42, C42, C4⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×3], Q16 [×4], C2×Q8, C2×Q8 [×2], C4×C8, C8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4⋊C8, C4.Q8 [×2], C4×Q8, C4⋊Q8 [×2], C2×Q16 [×2], C4.10D8 [×2], C8⋊2C8, C8⋊4Q8, Q8⋊Q8 [×2], C4⋊Q16, C8.SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C8⋊C22, C8.C22 [×2], 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 |
(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 86 103 33 18 119 52 47)(2 81 104 36 19 114 53 42)(3 84 97 39 20 117 54 45)(4 87 98 34 21 120 55 48)(5 82 99 37 22 115 56 43)(6 85 100 40 23 118 49 46)(7 88 101 35 24 113 50 41)(8 83 102 38 17 116 51 44)(9 77 109 30 122 64 91 71)(10 80 110 25 123 59 92 66)(11 75 111 28 124 62 93 69)(12 78 112 31 125 57 94 72)(13 73 105 26 126 60 95 67)(14 76 106 29 127 63 96 70)(15 79 107 32 128 58 89 65)(16 74 108 27 121 61 90 68)
(1 106 5 110)(2 105 6 109)(3 112 7 108)(4 111 8 107)(9 104 13 100)(10 103 14 99)(11 102 15 98)(12 101 16 97)(17 89 21 93)(18 96 22 92)(19 95 23 91)(20 94 24 90)(25 37 29 33)(26 36 30 40)(27 35 31 39)(28 34 32 38)(41 72 45 68)(42 71 46 67)(43 70 47 66)(44 69 48 65)(49 122 53 126)(50 121 54 125)(51 128 55 124)(52 127 56 123)(57 84 61 88)(58 83 62 87)(59 82 63 86)(60 81 64 85)(73 114 77 118)(74 113 78 117)(75 120 79 116)(76 119 80 115)
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), (1,86,103,33,18,119,52,47)(2,81,104,36,19,114,53,42)(3,84,97,39,20,117,54,45)(4,87,98,34,21,120,55,48)(5,82,99,37,22,115,56,43)(6,85,100,40,23,118,49,46)(7,88,101,35,24,113,50,41)(8,83,102,38,17,116,51,44)(9,77,109,30,122,64,91,71)(10,80,110,25,123,59,92,66)(11,75,111,28,124,62,93,69)(12,78,112,31,125,57,94,72)(13,73,105,26,126,60,95,67)(14,76,106,29,127,63,96,70)(15,79,107,32,128,58,89,65)(16,74,108,27,121,61,90,68), (1,106,5,110)(2,105,6,109)(3,112,7,108)(4,111,8,107)(9,104,13,100)(10,103,14,99)(11,102,15,98)(12,101,16,97)(17,89,21,93)(18,96,22,92)(19,95,23,91)(20,94,24,90)(25,37,29,33)(26,36,30,40)(27,35,31,39)(28,34,32,38)(41,72,45,68)(42,71,46,67)(43,70,47,66)(44,69,48,65)(49,122,53,126)(50,121,54,125)(51,128,55,124)(52,127,56,123)(57,84,61,88)(58,83,62,87)(59,82,63,86)(60,81,64,85)(73,114,77,118)(74,113,78,117)(75,120,79,116)(76,119,80,115)>;
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), (1,86,103,33,18,119,52,47)(2,81,104,36,19,114,53,42)(3,84,97,39,20,117,54,45)(4,87,98,34,21,120,55,48)(5,82,99,37,22,115,56,43)(6,85,100,40,23,118,49,46)(7,88,101,35,24,113,50,41)(8,83,102,38,17,116,51,44)(9,77,109,30,122,64,91,71)(10,80,110,25,123,59,92,66)(11,75,111,28,124,62,93,69)(12,78,112,31,125,57,94,72)(13,73,105,26,126,60,95,67)(14,76,106,29,127,63,96,70)(15,79,107,32,128,58,89,65)(16,74,108,27,121,61,90,68), (1,106,5,110)(2,105,6,109)(3,112,7,108)(4,111,8,107)(9,104,13,100)(10,103,14,99)(11,102,15,98)(12,101,16,97)(17,89,21,93)(18,96,22,92)(19,95,23,91)(20,94,24,90)(25,37,29,33)(26,36,30,40)(27,35,31,39)(28,34,32,38)(41,72,45,68)(42,71,46,67)(43,70,47,66)(44,69,48,65)(49,122,53,126)(50,121,54,125)(51,128,55,124)(52,127,56,123)(57,84,61,88)(58,83,62,87)(59,82,63,86)(60,81,64,85)(73,114,77,118)(74,113,78,117)(75,120,79,116)(76,119,80,115) );
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)], [(1,86,103,33,18,119,52,47),(2,81,104,36,19,114,53,42),(3,84,97,39,20,117,54,45),(4,87,98,34,21,120,55,48),(5,82,99,37,22,115,56,43),(6,85,100,40,23,118,49,46),(7,88,101,35,24,113,50,41),(8,83,102,38,17,116,51,44),(9,77,109,30,122,64,91,71),(10,80,110,25,123,59,92,66),(11,75,111,28,124,62,93,69),(12,78,112,31,125,57,94,72),(13,73,105,26,126,60,95,67),(14,76,106,29,127,63,96,70),(15,79,107,32,128,58,89,65),(16,74,108,27,121,61,90,68)], [(1,106,5,110),(2,105,6,109),(3,112,7,108),(4,111,8,107),(9,104,13,100),(10,103,14,99),(11,102,15,98),(12,101,16,97),(17,89,21,93),(18,96,22,92),(19,95,23,91),(20,94,24,90),(25,37,29,33),(26,36,30,40),(27,35,31,39),(28,34,32,38),(41,72,45,68),(42,71,46,67),(43,70,47,66),(44,69,48,65),(49,122,53,126),(50,121,54,125),(51,128,55,124),(52,127,56,123),(57,84,61,88),(58,83,62,87),(59,82,63,86),(60,81,64,85),(73,114,77,118),(74,113,78,117),(75,120,79,116),(76,119,80,115)])
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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