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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

Aliases: Q8.6SD16, C42.214C23, C4⋊C4.36D4, Q8⋊C8.7C2, (C2×Q8).54D4, C83Q8.4C2, C4.62(C4○D8), C4⋊C8.17C22, Q83Q8.3C2, Q8⋊Q8.5C2, C42Q16.3C2, C4.36(C2×SD16), C4⋊Q8.34C22, (C4×C8).247C22, C4.10D8.6C2, (C4×Q8).42C22, C2.17(Q8⋊D4), C4.69(C8.C22), C22.180C22≀C2, C2.26(D4.7D4), C2.18(D4.10D4), (C2×C4).971(C2×D4), SmallGroup(128,385)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — Q8.SD16
C1C2C22C2×C4C42C4×Q8Q83Q8 — Q8.SD16
C1C22C42 — Q8.SD16
C1C22C42 — Q8.SD16
C1C22C22C42 — 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 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F8G8H
 size 111122224444488881644448888
ρ111111111111111111111111111    trivial
ρ211111111-111-111-1-1-11-1-1-1-111-1-1    linear of order 2
ρ311111111-111-111-1-1-1-11111-1-111    linear of order 2
ρ411111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-11-1-1-1-11-1111111-1-1-1-1    linear of order 2
ρ61111111111-11-1-1-11-11-1-1-1-1-1-111    linear of order 2
ρ71111111111-11-1-1-11-1-1111111-1-1    linear of order 2
ρ811111111-11-1-1-1-11-11-1-1-1-1-11111    linear of order 2
ρ92222-222-22-202000-20000000000    orthogonal lifted from D4
ρ102222-222-2-2-20-200020000000000    orthogonal lifted from D4
ρ112222-2-2-2-202000020-2000000000    orthogonal lifted from D4
ρ122222-2-2-2-2020000-202000000000    orthogonal lifted from D4
ρ1322222-2-220-2202-2000000000000    orthogonal lifted from D4
ρ1422222-2-220-2-20-22000000000000    orthogonal lifted from D4
ρ1522-2-202-20-2002000000--2-2-2--2-2--200    complex lifted from SD16
ρ162-2-22200-200-2i02i00000-2-2--2--2002-2    complex lifted from C4○D8
ρ1722-2-202-20200-2000000-2--2--2-2-2--200    complex lifted from SD16
ρ182-2-22200-2002i0-2i00000-2-2--2--200-22    complex lifted from C4○D8
ρ1922-2-202-20-2002000000-2--2--2-2--2-200    complex lifted from SD16
ρ202-2-22200-2002i0-2i00000--2--2-2-2002-2    complex lifted from C4○D8
ρ2122-2-202-20200-2000000--2-2-2--2--2-200    complex lifted from SD16
ρ222-2-22200-200-2i02i00000--2--2-2-200-22    complex lifted from C4○D8
ρ234-44-400000000000000-22-220000    symplectic lifted from D4.10D4, Schur index 2
ρ244-44-4000000000000002-22-20000    symplectic lifted from D4.10D4, Schur index 2
ρ2544-4-40-440000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4-44-4004000000000000000000    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

1000
0100
0001
00160
,
16000
01600
00110
001016
,
121200
51200
001212
00512
,
16000
0100
0017
00716
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

Export

Character table of Q8.SD16 in TeX

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