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## G = Q8×Q16order 128 = 27

### Direct product of Q8 and Q16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — Q8×Q16
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — Q82 — Q8×Q16
 Lower central C1 — C2 — C2×C4 — Q8×Q16
 Upper central C1 — C22 — C4×Q8 — Q8×Q16
 Jennings C1 — C2 — C2 — C2×C4 — Q8×Q16

Generators and relations for Q8×Q16
G = < a,b,c,d | a4=c8=1, b2=a2, d2=c4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 280 in 168 conjugacy classes, 104 normal (14 characteristic)
C1, C2, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4×Q8, C4×Q8, C4⋊Q8, C4⋊Q8, C2×Q16, C4×Q16, C8×Q8, C4.Q16, C82Q8, Q82, Q8×Q16
Quotients: C1, C2, C22, D4, Q8, C23, Q16, C2×D4, C2×Q8, C24, C2×Q16, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C22×Q16, D4○D8, Q8×Q16

Smallest permutation representation of Q8×Q16
Regular action on 128 points
Generators in S128
(1 22 43 111)(2 23 44 112)(3 24 45 105)(4 17 46 106)(5 18 47 107)(6 19 48 108)(7 20 41 109)(8 21 42 110)(9 82 115 35)(10 83 116 36)(11 84 117 37)(12 85 118 38)(13 86 119 39)(14 87 120 40)(15 88 113 33)(16 81 114 34)(25 78 53 102)(26 79 54 103)(27 80 55 104)(28 73 56 97)(29 74 49 98)(30 75 50 99)(31 76 51 100)(32 77 52 101)(57 122 89 68)(58 123 90 69)(59 124 91 70)(60 125 92 71)(61 126 93 72)(62 127 94 65)(63 128 95 66)(64 121 96 67)
(1 58 43 90)(2 59 44 91)(3 60 45 92)(4 61 46 93)(5 62 47 94)(6 63 48 95)(7 64 41 96)(8 57 42 89)(9 103 115 79)(10 104 116 80)(11 97 117 73)(12 98 118 74)(13 99 119 75)(14 100 120 76)(15 101 113 77)(16 102 114 78)(17 72 106 126)(18 65 107 127)(19 66 108 128)(20 67 109 121)(21 68 110 122)(22 69 111 123)(23 70 112 124)(24 71 105 125)(25 81 53 34)(26 82 54 35)(27 83 55 36)(28 84 56 37)(29 85 49 38)(30 86 50 39)(31 87 51 40)(32 88 52 33)
(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 54 5 50)(2 53 6 49)(3 52 7 56)(4 51 8 55)(9 127 13 123)(10 126 14 122)(11 125 15 121)(12 124 16 128)(17 100 21 104)(18 99 22 103)(19 98 23 102)(20 97 24 101)(25 48 29 44)(26 47 30 43)(27 46 31 42)(28 45 32 41)(33 64 37 60)(34 63 38 59)(35 62 39 58)(36 61 40 57)(65 119 69 115)(66 118 70 114)(67 117 71 113)(68 116 72 120)(73 105 77 109)(74 112 78 108)(75 111 79 107)(76 110 80 106)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4H 4I ··· 4O 4P ··· 4U 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 ··· 4 8 ··· 8 2 2 2 2 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + - + - - + image C1 C2 C2 C2 C2 C2 D4 Q8 D4 Q16 2- 1+4 D4○D8 kernel Q8×Q16 C4×Q16 C8×Q8 C4.Q16 C8⋊2Q8 Q82 C4⋊C4 Q16 C2×Q8 Q8 C4 C2 # reps 1 3 1 6 3 2 3 4 1 8 1 2

Matrix representation of Q8×Q16 in GL4(𝔽17) generated by

 0 16 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 5 12 0 0 12 12 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 0 11 0 0 3 11
,
 16 0 0 0 0 16 0 0 0 0 0 7 0 0 12 0
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[5,12,0,0,12,12,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,16,0,0,0,0,0,12,0,0,7,0] >;

Q8×Q16 in GAP, Magma, Sage, TeX

Q_8\times Q_{16}
% in TeX

G:=Group("Q8xQ16");
// GroupNames label

G:=SmallGroup(128,2114);
// by ID

G=gap.SmallGroup(128,2114);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,352,346,80,4037,1027,124]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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