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G = Q86Q16order 128 = 27

2nd semidirect product of Q8 and Q16 acting through Inn(Q8)

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

Aliases: Q86Q16, C42.529C23, C4.1462+ 1+4, C4⋊C4.287D4, Q83(C2.D8), (C8×Q8).10C2, C4.32(C2×Q16), C8.95(C4○D4), (C4×C8).98C22, (C4×Q16).11C2, Q83Q8.9C2, (C2×Q8).275D4, C2.71(Q8○D8), C4⋊C8.307C22, C4⋊C4.446C23, (C2×C8).220C23, (C2×C4).587C24, C4⋊Q16.11C2, C42Q16.11C2, C4⋊Q8.215C22, C2.23(C22×Q16), C2.41(Q86D4), (C2×Q8).266C23, (C4×Q8).316C22, (C2×Q16).41C22, C2.D8.238C22, C22.847(C22×D4), Q8⋊C4.167C22, C4.165(C2×C4○D4), (C2×C4).1107(C2×D4), SmallGroup(128,2127)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q86Q16
C1C2C4C2×C4C42C4×Q8Q83Q8 — Q86Q16
C1C2C2×C4 — Q86Q16
C1C22C4×Q8 — Q86Q16
C1C2C2C2×C4 — Q86Q16

Generators and relations for Q86Q16
 G = < a,b,c,d | a4=c8=1, b2=a2, d2=c4, bab-1=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c-1 >

Subgroups: 280 in 174 conjugacy classes, 96 normal (14 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C4 [×11], C22, C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×12], C42 [×3], C42 [×6], C4⋊C4 [×5], C4⋊C4 [×18], C2×C8, C2×C8 [×3], Q16 [×12], C2×Q8, C2×Q8 [×6], C4×C8 [×3], Q8⋊C4 [×6], C4⋊C8 [×3], C2.D8, C4×Q8, C4×Q8 [×6], C4×Q8 [×2], C42.C2 [×6], C4⋊Q8 [×6], C2×Q16 [×9], C4×Q16 [×3], C8×Q8, C42Q16 [×6], C4⋊Q16 [×3], Q83Q8 [×2], Q86Q16
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, Q86D4, C22×Q16, Q8○D8, Q86Q16

Smallest permutation representation of Q86Q16
Regular action on 128 points
Generators in S128
(1 127 51 115)(2 116 52 128)(3 121 53 117)(4 118 54 122)(5 123 55 119)(6 120 56 124)(7 125 49 113)(8 114 50 126)(9 69 57 110)(10 111 58 70)(11 71 59 112)(12 105 60 72)(13 65 61 106)(14 107 62 66)(15 67 63 108)(16 109 64 68)(17 42 86 89)(18 90 87 43)(19 44 88 91)(20 92 81 45)(21 46 82 93)(22 94 83 47)(23 48 84 95)(24 96 85 41)(25 76 35 97)(26 98 36 77)(27 78 37 99)(28 100 38 79)(29 80 39 101)(30 102 40 73)(31 74 33 103)(32 104 34 75)
(1 99 51 78)(2 79 52 100)(3 101 53 80)(4 73 54 102)(5 103 55 74)(6 75 56 104)(7 97 49 76)(8 77 50 98)(9 87 57 18)(10 19 58 88)(11 81 59 20)(12 21 60 82)(13 83 61 22)(14 23 62 84)(15 85 63 24)(16 17 64 86)(25 125 35 113)(26 114 36 126)(27 127 37 115)(28 116 38 128)(29 121 39 117)(30 118 40 122)(31 123 33 119)(32 120 34 124)(41 67 96 108)(42 109 89 68)(43 69 90 110)(44 111 91 70)(45 71 92 112)(46 105 93 72)(47 65 94 106)(48 107 95 66)
(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 95 5 91)(2 94 6 90)(3 93 7 89)(4 92 8 96)(9 38 13 34)(10 37 14 33)(11 36 15 40)(12 35 16 39)(17 117 21 113)(18 116 22 120)(19 115 23 119)(20 114 24 118)(25 64 29 60)(26 63 30 59)(27 62 31 58)(28 61 32 57)(41 54 45 50)(42 53 46 49)(43 52 47 56)(44 51 48 55)(65 104 69 100)(66 103 70 99)(67 102 71 98)(68 101 72 97)(73 112 77 108)(74 111 78 107)(75 110 79 106)(76 109 80 105)(81 126 85 122)(82 125 86 121)(83 124 87 128)(84 123 88 127)

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

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

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

35 conjugacy classes

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

35 irreducible representations

dim111111222244
type++++++++-+-
imageC1C2C2C2C2C2D4D4C4○D4Q162+ 1+4Q8○D8
kernelQ86Q16C4×Q16C8×Q8C42Q16C4⋊Q16Q83Q8C4⋊C4C2×Q8C8Q8C4C2
# reps131632314812

Matrix representation of Q86Q16 in GL4(𝔽17) generated by

0100
16000
00160
00016
,
4000
01300
0010
0001
,
01300
13000
0006
00146
,
1000
01600
0062
00711
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,13,0,0,13,0,0,0,0,0,0,14,0,0,6,6],[1,0,0,0,0,16,0,0,0,0,6,7,0,0,2,11] >;

Q86Q16 in GAP, Magma, Sage, TeX

Q_8\rtimes_6Q_{16}
% in TeX

G:=Group("Q8:6Q16");
// GroupNames label

G:=SmallGroup(128,2127);
// by ID

G=gap.SmallGroup(128,2127);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,436,346,80,4037,1027,124]);
// Polycyclic

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

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