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## G = Q8.3SD16order 128 = 27

### 3rd non-split extension by Q8 of SD16 acting via SD16/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — Q8.3SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C8×Q8 — Q8.3SD16
 Lower central C1 — C22 — C42 — Q8.3SD16
 Upper central C1 — C22 — C42 — Q8.3SD16
 Jennings C1 — C22 — C22 — C42 — Q8.3SD16

Generators and relations for Q8.3SD16
G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c3 >

Subgroups: 152 in 75 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, Q8, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C4×C8, C4×C8, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C2×Q16, C4.10D8, C82C8, C8×Q8, C42Q16, Q8⋊Q8, C4.SD16, Q8.3SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C4○D8, C8.C22, Q8.D4, C88D4, D4.5D4, Q8.3SD16

Character table of Q8.3SD16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N size 1 1 1 1 2 2 2 2 4 4 4 4 4 16 16 2 2 2 2 4 4 4 4 4 4 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 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 -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 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 -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 -1 1 -1 linear of order 2 ρ9 2 2 2 2 2 -2 2 -2 0 0 0 -2 0 0 0 -2 -2 -2 -2 0 0 0 2 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 2 -2 2 2 -2 -2 -2 2 0 0 0 0 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 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 -2 2 -2 0 0 0 -2 0 0 0 2 2 2 2 0 0 0 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 -2 -2 0 0 0 2 0 0 0 0 0 0 0 2i 2i -2i 0 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ14 2 2 2 2 -2 -2 -2 -2 0 0 0 2 0 0 0 0 0 0 0 -2i -2i 2i 0 0 2i 0 0 0 0 complex lifted from C4○D4 ρ15 2 2 -2 -2 0 2 0 -2 0 2 -2 0 0 0 0 -√-2 √-2 √-2 -√-2 -√-2 √-2 √-2 √-2 -√-2 -√-2 0 0 0 0 complex lifted from SD16 ρ16 2 2 -2 -2 0 2 0 -2 0 -2 2 0 0 0 0 √-2 -√-2 -√-2 √-2 -√-2 √-2 √-2 -√-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ17 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 2i -2i 2i -2i 0 0 0 0 0 0 √2 -√-2 -√2 √-2 complex lifted from C4○D8 ρ18 2 2 -2 -2 0 -2 0 2 2i 0 0 0 -2i 0 0 -√-2 √-2 √-2 -√-2 √2 -√2 √2 -√-2 √-2 -√2 0 0 0 0 complex lifted from C4○D8 ρ19 2 2 -2 -2 0 -2 0 2 -2i 0 0 0 2i 0 0 √-2 -√-2 -√-2 √-2 √2 -√2 √2 √-2 -√-2 -√2 0 0 0 0 complex lifted from C4○D8 ρ20 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 -√2 -√-2 √2 √-2 complex lifted from C4○D8 ρ21 2 2 -2 -2 0 2 0 -2 0 -2 2 0 0 0 0 -√-2 √-2 √-2 -√-2 √-2 -√-2 -√-2 √-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ22 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 2i -2i 2i -2i 0 0 0 0 0 0 -√2 √-2 √2 -√-2 complex lifted from C4○D8 ρ23 2 2 -2 -2 0 2 0 -2 0 2 -2 0 0 0 0 √-2 -√-2 -√-2 √-2 √-2 -√-2 -√-2 -√-2 √-2 √-2 0 0 0 0 complex lifted from SD16 ρ24 2 2 -2 -2 0 -2 0 2 -2i 0 0 0 2i 0 0 -√-2 √-2 √-2 -√-2 -√2 √2 -√2 -√-2 √-2 √2 0 0 0 0 complex lifted from C4○D8 ρ25 2 2 -2 -2 0 -2 0 2 2i 0 0 0 -2i 0 0 √-2 -√-2 -√-2 √-2 -√2 √2 -√2 √-2 -√-2 √2 0 0 0 0 complex lifted from C4○D8 ρ26 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 √2 √-2 -√2 -√-2 complex lifted from C4○D8 ρ27 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 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ28 4 -4 -4 4 0 0 0 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 D4.5D4, Schur index 2 ρ29 4 -4 -4 4 0 0 0 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 D4.5D4, Schur index 2

Smallest permutation representation of Q8.3SD16
Regular action on 128 points
Generators in S128
(1 100 17 63)(2 101 18 64)(3 102 19 57)(4 103 20 58)(5 104 21 59)(6 97 22 60)(7 98 23 61)(8 99 24 62)(9 107 125 116)(10 108 126 117)(11 109 127 118)(12 110 128 119)(13 111 121 120)(14 112 122 113)(15 105 123 114)(16 106 124 115)(25 80 54 65)(26 73 55 66)(27 74 56 67)(28 75 49 68)(29 76 50 69)(30 77 51 70)(31 78 52 71)(32 79 53 72)(33 91 46 83)(34 92 47 84)(35 93 48 85)(36 94 41 86)(37 95 42 87)(38 96 43 88)(39 89 44 81)(40 90 45 82)
(1 26 17 55)(2 27 18 56)(3 28 19 49)(4 29 20 50)(5 30 21 51)(6 31 22 52)(7 32 23 53)(8 25 24 54)(9 89 125 81)(10 90 126 82)(11 91 127 83)(12 92 128 84)(13 93 121 85)(14 94 122 86)(15 95 123 87)(16 96 124 88)(33 118 46 109)(34 119 47 110)(35 120 48 111)(36 113 41 112)(37 114 42 105)(38 115 43 106)(39 116 44 107)(40 117 45 108)(57 75 102 68)(58 76 103 69)(59 77 104 70)(60 78 97 71)(61 79 98 72)(62 80 99 65)(63 73 100 66)(64 74 101 67)
(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)
(1 117 17 108)(2 111 18 120)(3 115 19 106)(4 109 20 118)(5 113 21 112)(6 107 22 116)(7 119 23 110)(8 105 24 114)(9 60 125 97)(10 100 126 63)(11 58 127 103)(12 98 128 61)(13 64 121 101)(14 104 122 59)(15 62 123 99)(16 102 124 57)(25 95 54 87)(26 82 55 90)(27 93 56 85)(28 88 49 96)(29 91 50 83)(30 86 51 94)(31 89 52 81)(32 84 53 92)(33 69 46 76)(34 79 47 72)(35 67 48 74)(36 77 41 70)(37 65 42 80)(38 75 43 68)(39 71 44 78)(40 73 45 66)

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

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

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

Matrix representation of Q8.3SD16 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 13 0 0 0 0 4
,
 16 0 0 0 0 16 0 0 0 0 0 1 0 0 16 0
,
 0 5 0 0 7 7 0 0 0 0 4 0 0 0 0 4
,
 0 14 0 0 11 0 0 0 0 0 0 9 0 0 15 0
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[0,7,0,0,5,7,0,0,0,0,4,0,0,0,0,4],[0,11,0,0,14,0,0,0,0,0,0,15,0,0,9,0] >;

Q8.3SD16 in GAP, Magma, Sage, TeX

Q_8._3{\rm SD}_{16}
% in TeX

G:=Group("Q8.3SD16");
// GroupNames label

G:=SmallGroup(128,412);
// by ID

G=gap.SmallGroup(128,412);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,352,1123,136,2804,718,172]);
// Polycyclic

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

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