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G = Q164Q8order 128 = 27

4th semidirect product of Q16 and Q8 acting via Q8/C4=C2

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

Aliases: Q164Q8, C42.68C23, C4.1022- 1+4, Q82.5C2, C8⋊Q8.2C2, C8.7(C2×Q8), C2.45(D4×Q8), C4⋊C4.390D4, Q8.12(C2×Q8), C84Q8.6C2, C83Q8.3C2, Q8.Q8.3C2, Q8⋊Q8.2C2, (C4×Q16).17C2, (C2×Q8).251D4, Q83Q8.7C2, C4.45(C22×Q8), C4⋊C8.146C22, C4⋊C4.276C23, (C4×C8).203C22, (C2×C4).579C24, (C2×C8).376C23, Q16⋊C4.2C2, C4.Q16.11C2, C4⋊Q8.208C22, C8⋊C4.72C22, C4.82(C8.C22), (C2×Q8).413C23, (C4×Q8).206C22, C4.Q8.119C22, C2.D8.140C22, C2.109(D4○SD16), Q8⋊C4.92C22, (C2×Q16).164C22, C22.839(C22×D4), C42.C2.77C22, (C2×C4).649(C2×D4), C2.91(C2×C8.C22), SmallGroup(128,2119)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q164Q8
C1C2C4C2×C4C42C4×Q8Q83Q8 — Q164Q8
C1C2C2×C4 — Q164Q8
C1C22C4×Q8 — Q164Q8
C1C2C2C2×C4 — Q164Q8

Generators and relations for Q164Q8
 G = < a,b,c,d | a8=c4=1, b2=a4, d2=c2, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 264 in 163 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×15], C22, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×9], C42, C42 [×2], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×14], C2×C8 [×2], C2×C8 [×2], Q16 [×4], C2×Q8 [×3], C2×Q8 [×4], C4×C8, C8⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×6], C2.D8, C2.D8 [×2], C4×Q8 [×3], C4×Q8 [×4], C4×Q8 [×2], C42.C2 [×2], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×2], C4⋊Q8 [×4], C2×Q16, C4×Q16, Q16⋊C4 [×2], C84Q8, Q8⋊Q8, Q8⋊Q8 [×2], C4.Q16, Q8.Q8 [×2], C83Q8, C8⋊Q8 [×2], Q83Q8, Q82, Q164Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C8.C22 [×2], C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C8.C22, D4○SD16, Q164Q8

Character table of Q164Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S8A8B8C8D8E8F
 size 11112222444444444888888444488
ρ111111111111111111111111111111    trivial
ρ21111-11-111-1-111-1-11-1111-1-1-1-1-111-11    linear of order 2
ρ31111-11-11-11-11-11-11-11-1-111-1-1-1111-1    linear of order 2
ρ411111111-1-111-1-11111-1-1-1-111111-1-1    linear of order 2
ρ511111111-1-11-1-1-1-1-1-1-11111-11111-1-1    linear of order 2
ρ61111-11-11-11-1-1-111-11-111-1-11-1-1111-1    linear of order 2
ρ71111-11-111-1-1-11-11-11-1-1-1111-1-111-11    linear of order 2
ρ811111111111-111-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ911111111-1-111-1-1-11-1-1-111-11-1-1-1-111    linear of order 2
ρ101111-11-11-11-11-11111-1-11-11-111-1-1-11    linear of order 2
ρ111111-11-111-1-111-1111-11-11-1-111-1-11-1    linear of order 2
ρ1211111111111111-11-1-11-1-111-1-1-1-1-1-1    linear of order 2
ρ1311111111111-1111-111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ141111-11-111-1-1-11-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ151111-11-11-11-1-1-11-1-1-111-11-1111-1-1-11    linear of order 2
ρ1611111111-1-11-1-1-11-1111-1-11-1-1-1-1-111    linear of order 2
ρ1722222-22-222-20-2-2000000000000000    orthogonal lifted from D4
ρ182222-2-2-2-22-220-22000000000000000    orthogonal lifted from D4
ρ1922222-22-2-2-2-2022000000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-22202-2000000000000000    orthogonal lifted from D4
ρ212-22-2020-2000-200-2220000002-20000    symplectic lifted from Q8, Schur index 2
ρ222-22-2020-20002002-2-20000002-20000    symplectic lifted from Q8, Schur index 2
ρ232-22-2020-2000-20022-2000000-220000    symplectic lifted from Q8, Schur index 2
ρ242-22-2020-2000200-2-22000000-220000    symplectic lifted from Q8, Schur index 2
ρ254-4-4440-40000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-44-40-404000000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ274-4-44-4040000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of Q164Q8
Regular action on 128 points
Generators in S128
(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 55 5 51)(2 54 6 50)(3 53 7 49)(4 52 8 56)(9 116 13 120)(10 115 14 119)(11 114 15 118)(12 113 16 117)(17 107 21 111)(18 106 22 110)(19 105 23 109)(20 112 24 108)(25 127 29 123)(26 126 30 122)(27 125 31 121)(28 124 32 128)(33 75 37 79)(34 74 38 78)(35 73 39 77)(36 80 40 76)(41 65 45 69)(42 72 46 68)(43 71 47 67)(44 70 48 66)(57 101 61 97)(58 100 62 104)(59 99 63 103)(60 98 64 102)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)
(1 59 42 107)(2 60 43 108)(3 61 44 109)(4 62 45 110)(5 63 46 111)(6 64 47 112)(7 57 48 105)(8 58 41 106)(9 73 126 89)(10 74 127 90)(11 75 128 91)(12 76 121 92)(13 77 122 93)(14 78 123 94)(15 79 124 95)(16 80 125 96)(17 51 103 68)(18 52 104 69)(19 53 97 70)(20 54 98 71)(21 55 99 72)(22 56 100 65)(23 49 101 66)(24 50 102 67)(25 86 119 34)(26 87 120 35)(27 88 113 36)(28 81 114 37)(29 82 115 38)(30 83 116 39)(31 84 117 40)(32 85 118 33)
(1 74 42 90)(2 79 43 95)(3 76 44 92)(4 73 45 89)(5 78 46 94)(6 75 47 91)(7 80 48 96)(8 77 41 93)(9 110 126 62)(10 107 127 59)(11 112 128 64)(12 109 121 61)(13 106 122 58)(14 111 123 63)(15 108 124 60)(16 105 125 57)(17 29 103 115)(18 26 104 120)(19 31 97 117)(20 28 98 114)(21 25 99 119)(22 30 100 116)(23 27 101 113)(24 32 102 118)(33 67 85 50)(34 72 86 55)(35 69 87 52)(36 66 88 49)(37 71 81 54)(38 68 82 51)(39 65 83 56)(40 70 84 53)

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

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

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

Matrix representation of Q164Q8 in GL6(𝔽17)

100000
010000
00125010
00512107
0012500
0012101210
,
1600000
0160000
0051134
004151010
004101616
005141215
,
010000
1600000
000010
00116115
0016000
0016101
,
400000
0130000
0034812
00161236
001691013
0036139

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,12,12,0,0,5,12,5,10,0,0,0,10,0,12,0,0,10,7,0,10],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,4,4,5,0,0,11,15,10,14,0,0,3,10,16,12,0,0,4,10,16,15],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,3,16,16,3,0,0,4,12,9,6,0,0,8,3,10,13,0,0,12,6,13,9] >;

Q164Q8 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_4Q_8
% in TeX

G:=Group("Q16:4Q8");
// GroupNames label

G:=SmallGroup(128,2119);
// by ID

G=gap.SmallGroup(128,2119);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,723,352,346,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^4=1,b^2=a^4,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of Q164Q8 in TeX

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