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G = Q85Q16order 128 = 27

1st semidirect product of Q8 and Q16 acting through Inn(Q8)

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

Aliases: Q85Q16, C42.504C23, C4.252- 1+4, Q82.3C2, (C8×Q8).8C2, C4⋊C4.277D4, (C4×Q16).8C2, C4.30(C2×Q16), Q83(Q8⋊C4), (C4×C8).93C22, (C2×Q8).269D4, C4.Q16.9C2, Q83Q8.4C2, C4⋊C4.431C23, C4⋊C8.303C22, (C2×C8).206C23, (C2×C4).555C24, Q8.33(C4○D4), C42Q16.10C2, C4⋊Q8.184C22, C4.SD16.8C2, C2.21(C22×Q16), C2.63(Q85D4), C2.98(D4○SD16), (C4×Q8).309C22, (C2×Q8).253C23, C2.D8.201C22, (C2×Q16).141C22, Q8⋊C4.18C22, C22.815(C22×D4), C4.256(C2×C4○D4), (C2×C4).1101(C2×D4), SmallGroup(128,2095)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q85Q16
C1C2C4C2×C4C42C4×Q8Q82 — Q85Q16
C1C2C2×C4 — Q85Q16
C1C22C4×Q8 — Q85Q16
C1C2C2C2×C4 — Q85Q16

Generators and relations for Q85Q16
 G = < a,b,c,d | a4=c8=1, b2=a2, d2=c4, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >

Subgroups: 272 in 168 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C4 [×12], C22, C8 [×4], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×6], Q8 [×10], C42 [×3], C42 [×6], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×15], C2×C8, C2×C8 [×3], Q16 [×6], C2×Q8 [×2], C2×Q8 [×3], C2×Q8 [×3], C4×C8 [×3], Q8⋊C4, Q8⋊C4 [×9], C4⋊C8 [×3], C2.D8 [×3], C4×Q8, C4×Q8 [×6], C4×Q8 [×2], C42.C2 [×3], C4⋊Q8 [×6], C4⋊Q8 [×3], C2×Q16 [×3], C4×Q16 [×3], C8×Q8, C42Q16 [×3], C4.Q16 [×3], C4.SD16 [×3], Q83Q8, Q82, Q85Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×Q16 [×6], C22×D4, C2×C4○D4, 2- 1+4, Q85D4, C22×Q16, D4○SD16, Q85Q16

Smallest permutation representation of Q85Q16
Regular action on 128 points
Generators in S128
(1 39 107 89)(2 40 108 90)(3 33 109 91)(4 34 110 92)(5 35 111 93)(6 36 112 94)(7 37 105 95)(8 38 106 96)(9 115 47 56)(10 116 48 49)(11 117 41 50)(12 118 42 51)(13 119 43 52)(14 120 44 53)(15 113 45 54)(16 114 46 55)(17 125 84 77)(18 126 85 78)(19 127 86 79)(20 128 87 80)(21 121 88 73)(22 122 81 74)(23 123 82 75)(24 124 83 76)(25 63 100 71)(26 64 101 72)(27 57 102 65)(28 58 103 66)(29 59 104 67)(30 60 97 68)(31 61 98 69)(32 62 99 70)
(1 127 107 79)(2 80 108 128)(3 121 109 73)(4 74 110 122)(5 123 111 75)(6 76 112 124)(7 125 105 77)(8 78 106 126)(9 66 47 58)(10 59 48 67)(11 68 41 60)(12 61 42 69)(13 70 43 62)(14 63 44 71)(15 72 45 64)(16 57 46 65)(17 95 84 37)(18 38 85 96)(19 89 86 39)(20 40 87 90)(21 91 88 33)(22 34 81 92)(23 93 82 35)(24 36 83 94)(25 53 100 120)(26 113 101 54)(27 55 102 114)(28 115 103 56)(29 49 104 116)(30 117 97 50)(31 51 98 118)(32 119 99 52)
(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 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 23 13 19)(10 22 14 18)(11 21 15 17)(12 20 16 24)(33 64 37 60)(34 63 38 59)(35 62 39 58)(36 61 40 57)(41 88 45 84)(42 87 46 83)(43 86 47 82)(44 85 48 81)(49 74 53 78)(50 73 54 77)(51 80 55 76)(52 79 56 75)(65 94 69 90)(66 93 70 89)(67 92 71 96)(68 91 72 95)(97 109 101 105)(98 108 102 112)(99 107 103 111)(100 106 104 110)(113 125 117 121)(114 124 118 128)(115 123 119 127)(116 122 120 126)

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

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

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

35 conjugacy classes

class 1 2A2B2C4A···4H4I···4O4P···4U8A8B8C8D8E···8J
order12224···44···44···488888···8
size11112···24···48···822224···4

35 irreducible representations

dim11111111222244
type++++++++++--
imageC1C2C2C2C2C2C2C2D4D4Q16C4○D42- 1+4D4○SD16
kernelQ85Q16C4×Q16C8×Q8C42Q16C4.Q16C4.SD16Q83Q8Q82C4⋊C4C2×Q8Q8Q8C4C2
# reps13133311318412

Matrix representation of Q85Q16 in GL4(𝔽17) generated by

16000
01600
0001
00160
,
16000
01600
0071
00110
,
8000
51500
00013
0040
,
11500
11600
00160
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,7,1,0,0,1,10],[8,5,0,0,0,15,0,0,0,0,0,4,0,0,13,0],[1,1,0,0,15,16,0,0,0,0,16,0,0,0,0,16] >;

Q85Q16 in GAP, Magma, Sage, TeX

Q_8\rtimes_5Q_{16}
% in TeX

G:=Group("Q8:5Q16");
// GroupNames label

G:=SmallGroup(128,2095);
// by ID

G=gap.SmallGroup(128,2095);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,352,346,80,4037,1027,124]);
// Polycyclic

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

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