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G = Q8.SD16order 128 = 27

2nd non-split extension by Q8 of SD16 acting via SD16/Q8=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8.SD16
G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=ab, dbd-1=a-1b, dcd-1=c3 >

Subgroups: 192 in 97 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, Q8, Q8, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, C4⋊Q8, C2×Q16, Q8⋊C8, C4.10D8, C42Q16, Q8⋊Q8, C83Q8, Q83Q8, Q8.SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C4○D8, C8.C22, Q8⋊D4, D4.7D4, D4.10D4, Q8.SD16

Character table of Q8.SD16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 2 2 4 4 4 4 4 8 8 8 8 16 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 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 2 0 0 0 -2 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 0 -2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -2 -2 0 2 0 0 0 0 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 -2 -2 0 2 0 0 0 0 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 -2 -2 2 0 -2 2 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 2 -2 -2 2 0 -2 -2 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 0 2 -2 0 -2 0 0 2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 √-2 -√-2 0 0 complex lifted from SD16 ρ16 2 -2 -2 2 2 0 0 -2 0 0 -2i 0 2i 0 0 0 0 0 √-2 √-2 -√-2 -√-2 0 0 √2 -√2 complex lifted from C4○D8 ρ17 2 2 -2 -2 0 2 -2 0 2 0 0 -2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 √-2 -√-2 0 0 complex lifted from SD16 ρ18 2 -2 -2 2 2 0 0 -2 0 0 2i 0 -2i 0 0 0 0 0 √-2 √-2 -√-2 -√-2 0 0 -√2 √2 complex lifted from C4○D8 ρ19 2 2 -2 -2 0 2 -2 0 -2 0 0 2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 -√-2 √-2 0 0 complex lifted from SD16 ρ20 2 -2 -2 2 2 0 0 -2 0 0 2i 0 -2i 0 0 0 0 0 -√-2 -√-2 √-2 √-2 0 0 √2 -√2 complex lifted from C4○D8 ρ21 2 2 -2 -2 0 2 -2 0 2 0 0 -2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 -√-2 √-2 0 0 complex lifted from SD16 ρ22 2 -2 -2 2 2 0 0 -2 0 0 -2i 0 2i 0 0 0 0 0 -√-2 -√-2 √-2 √-2 0 0 -√2 √2 complex lifted from C4○D8 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ25 4 4 -4 -4 0 -4 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 -4 0 0 4 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

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

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

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

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

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

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 16 0
,
 16 0 0 0 0 16 0 0 0 0 1 10 0 0 10 16
,
 12 12 0 0 5 12 0 0 0 0 12 12 0 0 5 12
,
 16 0 0 0 0 1 0 0 0 0 1 7 0 0 7 16
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,1,10,0,0,10,16],[12,5,0,0,12,12,0,0,0,0,12,5,0,0,12,12],[16,0,0,0,0,1,0,0,0,0,1,7,0,0,7,16] >;

Q8.SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(128,385);
// by ID

G=gap.SmallGroup(128,385);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,680,422,352,1123,570,521,136,2804,1411,718,172]);
// 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,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^3>;
// generators/relations

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