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G = Q8×Q16order 128 = 27

Direct product of Q8 and Q16

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

Aliases: Q8×Q16, C42.523C23, C4.972- 1+4, Q82.4C2, (C8×Q8).9C2, Q8.9(C2×Q8), C8.36(C2×Q8), C2.40(D4×Q8), C4⋊C4.282D4, (C4×Q16).9C2, C4.31(C2×Q16), C2.67(D4○D8), (C4×C8).96C22, (C2×Q8).272D4, C82Q8.16C2, C4.40(C22×Q8), C4⋊C4.271C23, C4⋊C8.305C22, (C2×C4).574C24, (C2×C8).212C23, C4.Q16.10C2, C4⋊Q8.203C22, C2.22(C22×Q16), C2.D8.73C22, (C2×Q8).409C23, (C4×Q8).202C22, (C2×Q16).172C22, C22.834(C22×D4), Q8⋊C4.164C22, (C2×C4).1104(C2×D4), SmallGroup(128,2114)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q8×Q16
C1C2C4C2×C4C42C4×Q8Q82 — Q8×Q16
C1C2C2×C4 — Q8×Q16
C1C22C4×Q8 — Q8×Q16
C1C2C2C2×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 [×3], C4 [×2], C4 [×6], C4 [×13], C22, C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×8], Q8 [×8], C42 [×3], C42 [×6], C4⋊C4 [×9], C4⋊C4 [×12], C2×C8, C2×C8 [×3], Q16 [×4], C2×Q8, C2×Q8 [×2], C2×Q8 [×6], C4×C8 [×3], Q8⋊C4 [×6], C4⋊C8 [×3], C2.D8 [×9], C4×Q8, C4×Q8 [×6], C4×Q8 [×2], C4⋊Q8 [×6], C4⋊Q8 [×6], C2×Q16, C4×Q16 [×3], C8×Q8, C4.Q16 [×6], C82Q8 [×3], Q82 [×2], Q8×Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C2×Q8 [×6], C24, C2×Q16 [×6], 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 25 43 111)(2 26 44 112)(3 27 45 105)(4 28 46 106)(5 29 47 107)(6 30 48 108)(7 31 41 109)(8 32 42 110)(9 84 117 37)(10 85 118 38)(11 86 119 39)(12 87 120 40)(13 88 113 33)(14 81 114 34)(15 82 115 35)(16 83 116 36)(17 54 103 72)(18 55 104 65)(19 56 97 66)(20 49 98 67)(21 50 99 68)(22 51 100 69)(23 52 101 70)(24 53 102 71)(57 77 122 89)(58 78 123 90)(59 79 124 91)(60 80 125 92)(61 73 126 93)(62 74 127 94)(63 75 128 95)(64 76 121 96)
(1 78 43 90)(2 79 44 91)(3 80 45 92)(4 73 46 93)(5 74 47 94)(6 75 48 95)(7 76 41 96)(8 77 42 89)(9 97 117 19)(10 98 118 20)(11 99 119 21)(12 100 120 22)(13 101 113 23)(14 102 114 24)(15 103 115 17)(16 104 116 18)(25 58 111 123)(26 59 112 124)(27 60 105 125)(28 61 106 126)(29 62 107 127)(30 63 108 128)(31 64 109 121)(32 57 110 122)(33 70 88 52)(34 71 81 53)(35 72 82 54)(36 65 83 55)(37 66 84 56)(38 67 85 49)(39 68 86 50)(40 69 87 51)
(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)(9 125 13 121)(10 124 14 128)(11 123 15 127)(12 122 16 126)(17 107 21 111)(18 106 22 110)(19 105 23 109)(20 112 24 108)(25 103 29 99)(26 102 30 98)(27 101 31 97)(28 100 32 104)(33 76 37 80)(34 75 38 79)(35 74 39 78)(36 73 40 77)(41 66 45 70)(42 65 46 69)(43 72 47 68)(44 71 48 67)(57 116 61 120)(58 115 62 119)(59 114 63 118)(60 113 64 117)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)

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

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

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

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+++++++-+--+
imageC1C2C2C2C2C2D4Q8D4Q162- 1+4D4○D8
kernelQ8×Q16C4×Q16C8×Q8C4.Q16C82Q8Q82C4⋊C4Q16C2×Q8Q8C4C2
# reps131632341812

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

01600
1000
0010
0001
,
51200
121200
00160
00016
,
1000
0100
00011
00311
,
16000
01600
0007
00120
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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