p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊6Q16, C42.529C23, C4.1462+ (1+4), C4⋊C4.287D4, Q8○3(C2.D8), (C8×Q8).10C2, C4.32(C2×Q16), C8.95(C4○D4), (C4×C8).98C22, Q8⋊3Q8.9C2, (C2×Q8).275D4, (C4×Q16).11C2, C2.71(Q8○D8), C4⋊C4.446C23, C4⋊C8.307C22, (C2×C4).587C24, (C2×C8).220C23, C4⋊Q16.11C2, C4⋊2Q16.11C2, C4⋊Q8.215C22, C2.23(C22×Q16), C2.41(Q8⋊6D4), (C2×Q16).41C22, (C4×Q8).316C22, (C2×Q8).266C23, 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
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, C4⋊2Q16 [×6], C4⋊Q16 [×3], Q8⋊3Q8 [×2], Q8⋊6Q16
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), Q8⋊6D4, C22×Q16, Q8○D8, Q8⋊6Q16
Generators and relations
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 >
►Regular action on 128 points
Generators in S128
(1 54 127 115)(2 116 128 55)(3 56 121 117)(4 118 122 49)(5 50 123 119)(6 120 124 51)(7 52 125 113)(8 114 126 53)(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 85 47 89)(18 90 48 86)(19 87 41 91)(20 92 42 88)(21 81 43 93)(22 94 44 82)(23 83 45 95)(24 96 46 84)(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 99 127 78)(2 79 128 100)(3 101 121 80)(4 73 122 102)(5 103 123 74)(6 75 124 104)(7 97 125 76)(8 77 126 98)(9 48 57 18)(10 19 58 41)(11 42 59 20)(12 21 60 43)(13 44 61 22)(14 23 62 45)(15 46 63 24)(16 17 64 47)(25 52 35 113)(26 114 36 53)(27 54 37 115)(28 116 38 55)(29 56 39 117)(30 118 40 49)(31 50 33 119)(32 120 34 51)(65 94 106 82)(66 83 107 95)(67 96 108 84)(68 85 109 89)(69 90 110 86)(70 87 111 91)(71 92 112 88)(72 81 105 93)
(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)(81 125 85 121)(82 124 86 128)(83 123 87 127)(84 122 88 126)
G:=sub<Sym(128)| (1,54,127,115)(2,116,128,55)(3,56,121,117)(4,118,122,49)(5,50,123,119)(6,120,124,51)(7,52,125,113)(8,114,126,53)(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)(17,85,47,89)(18,90,48,86)(19,87,41,91)(20,92,42,88)(21,81,43,93)(22,94,44,82)(23,83,45,95)(24,96,46,84)(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), (1,99,127,78)(2,79,128,100)(3,101,121,80)(4,73,122,102)(5,103,123,74)(6,75,124,104)(7,97,125,76)(8,77,126,98)(9,48,57,18)(10,19,58,41)(11,42,59,20)(12,21,60,43)(13,44,61,22)(14,23,62,45)(15,46,63,24)(16,17,64,47)(25,52,35,113)(26,114,36,53)(27,54,37,115)(28,116,38,55)(29,56,39,117)(30,118,40,49)(31,50,33,119)(32,120,34,51)(65,94,106,82)(66,83,107,95)(67,96,108,84)(68,85,109,89)(69,90,110,86)(70,87,111,91)(71,92,112,88)(72,81,105,93), (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)(81,125,85,121)(82,124,86,128)(83,123,87,127)(84,122,88,126)>;
G:=Group( (1,54,127,115)(2,116,128,55)(3,56,121,117)(4,118,122,49)(5,50,123,119)(6,120,124,51)(7,52,125,113)(8,114,126,53)(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)(17,85,47,89)(18,90,48,86)(19,87,41,91)(20,92,42,88)(21,81,43,93)(22,94,44,82)(23,83,45,95)(24,96,46,84)(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), (1,99,127,78)(2,79,128,100)(3,101,121,80)(4,73,122,102)(5,103,123,74)(6,75,124,104)(7,97,125,76)(8,77,126,98)(9,48,57,18)(10,19,58,41)(11,42,59,20)(12,21,60,43)(13,44,61,22)(14,23,62,45)(15,46,63,24)(16,17,64,47)(25,52,35,113)(26,114,36,53)(27,54,37,115)(28,116,38,55)(29,56,39,117)(30,118,40,49)(31,50,33,119)(32,120,34,51)(65,94,106,82)(66,83,107,95)(67,96,108,84)(68,85,109,89)(69,90,110,86)(70,87,111,91)(71,92,112,88)(72,81,105,93), (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)(81,125,85,121)(82,124,86,128)(83,123,87,127)(84,122,88,126) );
G=PermutationGroup([(1,54,127,115),(2,116,128,55),(3,56,121,117),(4,118,122,49),(5,50,123,119),(6,120,124,51),(7,52,125,113),(8,114,126,53),(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),(17,85,47,89),(18,90,48,86),(19,87,41,91),(20,92,42,88),(21,81,43,93),(22,94,44,82),(23,83,45,95),(24,96,46,84),(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)], [(1,99,127,78),(2,79,128,100),(3,101,121,80),(4,73,122,102),(5,103,123,74),(6,75,124,104),(7,97,125,76),(8,77,126,98),(9,48,57,18),(10,19,58,41),(11,42,59,20),(12,21,60,43),(13,44,61,22),(14,23,62,45),(15,46,63,24),(16,17,64,47),(25,52,35,113),(26,114,36,53),(27,54,37,115),(28,116,38,55),(29,56,39,117),(30,118,40,49),(31,50,33,119),(32,120,34,51),(65,94,106,82),(66,83,107,95),(67,96,108,84),(68,85,109,89),(69,90,110,86),(70,87,111,91),(71,92,112,88),(72,81,105,93)], [(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),(81,125,85,121),(82,124,86,128),(83,123,87,127),(84,122,88,126)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 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 | 6 |
| 0 | 0 | 14 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 6 | 2 |
| 0 | 0 | 7 | 11 |
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] >;
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 | D4 | C4○D4 | Q16 | 2+ (1+4) | Q8○D8 |
| kernel | Q8⋊6Q16 | C4×Q16 | C8×Q8 | C4⋊2Q16 | C4⋊Q16 | Q8⋊3Q8 | C4⋊C4 | C2×Q8 | C8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 3 | 1 | 4 | 8 | 1 | 2 |
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
×
⋊
ℤ
𝔽
○
≀