p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.14SD16, C42.155D4, (C2×C4).47D8, (C2×C8).137D4, C8.55(C4○D4), C16⋊5C4.10C2, C4.19(C2×SD16), C8.5Q8.6C2, (C2×C16).60C22, (C4×C8).169C22, (C2×C8).543C23, C2.Q32.7C2, C4⋊Q16.15C2, C22.129(C2×D8), C2.D8.28C22, C4.12(C4.4D4), C2.21(Q32⋊C2), C2.14(C4.4D8), (C2×Q16).16C22, (C2×C4).811(C2×D4), SmallGroup(128,977)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.14SD16
G = < a,b,c | a8=1, b8=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=a6b3 >
Subgroups: 152 in 66 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C16, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C4×C8, C4.Q8, C2.D8, C2×C16, C42.C2, C4⋊Q8, C2×Q16, C2×Q16, C16⋊5C4, C2.Q32, C4⋊Q16, C8.5Q8, C8.14SD16
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, C4.4D4, C2×D8, C2×SD16, C4.4D8, Q32⋊C2, C8.14SD16
Character table of C8.14SD16
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | -√2 | √2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 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 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 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 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | 0 | -2i | 2i | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ16 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | -2i | -2i | 2i | complex lifted from C4○D4 |
| ρ17 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 2i | 2i | -2i | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | 0 | 2i | -2i | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -√-2 | √-2 | √-2 | √-2 | -√-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ22 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | √-2 | -√-2 | -√-2 | -√-2 | √-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 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 | symplectic lifted from Q32⋊C2, Schur index 2 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 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 | symplectic lifted from Q32⋊C2, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 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 | symplectic lifted from Q32⋊C2, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 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 | symplectic lifted from Q32⋊C2, Schur index 2 |
►Regular action on 128 points
Generators in S128
(1 84 59 98 9 92 51 106)(2 93 60 107 10 85 52 99)(3 86 61 100 11 94 53 108)(4 95 62 109 12 87 54 101)(5 88 63 102 13 96 55 110)(6 81 64 111 14 89 56 103)(7 90 49 104 15 82 57 112)(8 83 50 97 16 91 58 105)(17 78 128 46 25 70 120 38)(18 71 113 39 26 79 121 47)(19 80 114 48 27 72 122 40)(20 73 115 41 28 65 123 33)(21 66 116 34 29 74 124 42)(22 75 117 43 30 67 125 35)(23 68 118 36 31 76 126 44)(24 77 119 45 32 69 127 37)
(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)
(1 120 9 128)(2 28 10 20)(3 118 11 126)(4 26 12 18)(5 116 13 124)(6 24 14 32)(7 114 15 122)(8 22 16 30)(17 59 25 51)(19 57 27 49)(21 55 29 63)(23 53 31 61)(33 93 41 85)(34 102 42 110)(35 91 43 83)(36 100 44 108)(37 89 45 81)(38 98 46 106)(39 87 47 95)(40 112 48 104)(50 125 58 117)(52 123 60 115)(54 121 62 113)(56 119 64 127)(65 107 73 99)(66 96 74 88)(67 105 75 97)(68 94 76 86)(69 103 77 111)(70 92 78 84)(71 101 79 109)(72 90 80 82)
G:=sub<Sym(128)| (1,84,59,98,9,92,51,106)(2,93,60,107,10,85,52,99)(3,86,61,100,11,94,53,108)(4,95,62,109,12,87,54,101)(5,88,63,102,13,96,55,110)(6,81,64,111,14,89,56,103)(7,90,49,104,15,82,57,112)(8,83,50,97,16,91,58,105)(17,78,128,46,25,70,120,38)(18,71,113,39,26,79,121,47)(19,80,114,48,27,72,122,40)(20,73,115,41,28,65,123,33)(21,66,116,34,29,74,124,42)(22,75,117,43,30,67,125,35)(23,68,118,36,31,76,126,44)(24,77,119,45,32,69,127,37), (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), (1,120,9,128)(2,28,10,20)(3,118,11,126)(4,26,12,18)(5,116,13,124)(6,24,14,32)(7,114,15,122)(8,22,16,30)(17,59,25,51)(19,57,27,49)(21,55,29,63)(23,53,31,61)(33,93,41,85)(34,102,42,110)(35,91,43,83)(36,100,44,108)(37,89,45,81)(38,98,46,106)(39,87,47,95)(40,112,48,104)(50,125,58,117)(52,123,60,115)(54,121,62,113)(56,119,64,127)(65,107,73,99)(66,96,74,88)(67,105,75,97)(68,94,76,86)(69,103,77,111)(70,92,78,84)(71,101,79,109)(72,90,80,82)>;
G:=Group( (1,84,59,98,9,92,51,106)(2,93,60,107,10,85,52,99)(3,86,61,100,11,94,53,108)(4,95,62,109,12,87,54,101)(5,88,63,102,13,96,55,110)(6,81,64,111,14,89,56,103)(7,90,49,104,15,82,57,112)(8,83,50,97,16,91,58,105)(17,78,128,46,25,70,120,38)(18,71,113,39,26,79,121,47)(19,80,114,48,27,72,122,40)(20,73,115,41,28,65,123,33)(21,66,116,34,29,74,124,42)(22,75,117,43,30,67,125,35)(23,68,118,36,31,76,126,44)(24,77,119,45,32,69,127,37), (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), (1,120,9,128)(2,28,10,20)(3,118,11,126)(4,26,12,18)(5,116,13,124)(6,24,14,32)(7,114,15,122)(8,22,16,30)(17,59,25,51)(19,57,27,49)(21,55,29,63)(23,53,31,61)(33,93,41,85)(34,102,42,110)(35,91,43,83)(36,100,44,108)(37,89,45,81)(38,98,46,106)(39,87,47,95)(40,112,48,104)(50,125,58,117)(52,123,60,115)(54,121,62,113)(56,119,64,127)(65,107,73,99)(66,96,74,88)(67,105,75,97)(68,94,76,86)(69,103,77,111)(70,92,78,84)(71,101,79,109)(72,90,80,82) );
G=PermutationGroup([[(1,84,59,98,9,92,51,106),(2,93,60,107,10,85,52,99),(3,86,61,100,11,94,53,108),(4,95,62,109,12,87,54,101),(5,88,63,102,13,96,55,110),(6,81,64,111,14,89,56,103),(7,90,49,104,15,82,57,112),(8,83,50,97,16,91,58,105),(17,78,128,46,25,70,120,38),(18,71,113,39,26,79,121,47),(19,80,114,48,27,72,122,40),(20,73,115,41,28,65,123,33),(21,66,116,34,29,74,124,42),(22,75,117,43,30,67,125,35),(23,68,118,36,31,76,126,44),(24,77,119,45,32,69,127,37)], [(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)], [(1,120,9,128),(2,28,10,20),(3,118,11,126),(4,26,12,18),(5,116,13,124),(6,24,14,32),(7,114,15,122),(8,22,16,30),(17,59,25,51),(19,57,27,49),(21,55,29,63),(23,53,31,61),(33,93,41,85),(34,102,42,110),(35,91,43,83),(36,100,44,108),(37,89,45,81),(38,98,46,106),(39,87,47,95),(40,112,48,104),(50,125,58,117),(52,123,60,115),(54,121,62,113),(56,119,64,127),(65,107,73,99),(66,96,74,88),(67,105,75,97),(68,94,76,86),(69,103,77,111),(70,92,78,84),(71,101,79,109),(72,90,80,82)]])
Matrix representation of C8.14SD16 ►in GL6(𝔽17)
| 6 | 9 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 13 | 9 | 3 |
| 0 | 0 | 10 | 6 | 7 | 1 |
| 0 | 0 | 0 | 0 | 14 | 9 |
| 0 | 0 | 1 | 5 | 8 | 3 |
| 7 | 2 | 0 | 0 | 0 | 0 |
| 10 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 4 | 0 | 15 |
| 0 | 0 | 11 | 15 | 10 | 5 |
| 0 | 0 | 8 | 15 | 9 | 11 |
| 0 | 0 | 9 | 0 | 3 | 15 |
| 11 | 14 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 2 | 16 |
| 0 | 0 | 9 | 14 | 7 | 7 |
| 0 | 0 | 2 | 6 | 15 | 11 |
| 0 | 0 | 9 | 12 | 0 | 3 |
G:=sub<GL(6,GF(17))| [6,11,0,0,0,0,9,11,0,0,0,0,0,0,11,10,0,1,0,0,13,6,0,5,0,0,9,7,14,8,0,0,3,1,9,3],[7,10,0,0,0,0,2,10,0,0,0,0,0,0,12,11,8,9,0,0,4,15,15,0,0,0,0,10,9,3,0,0,15,5,11,15],[11,6,0,0,0,0,14,6,0,0,0,0,0,0,2,9,2,9,0,0,0,14,6,12,0,0,2,7,15,0,0,0,16,7,11,3] >;
C8.14SD16 in GAP, Magma, Sage, TeX
C_8._{14}{\rm SD}_{16} % in TeX
G:=Group("C8.14SD16"); // GroupNames label
G:=SmallGroup(128,977);
// by ID
G=gap.SmallGroup(128,977);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,568,422,723,58,1123,360,3924,102,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^8=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^3>;
// generators/relations
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