p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊6Q8, C42.524C23, C4.982- 1+4, C4⋊C4○3Q16, C2.41(D4×Q8), C8.37(C2×Q8), C4⋊C4.417D4, (C8×Q8).11C2, Q8.10(C2×Q8), Q8.Q8.1C2, C4.93(C4○D8), Q8⋊3Q8.6C2, (C4×Q16).10C2, (C2×Q8).188D4, C2.67(Q8○D8), C8⋊2Q8.17C2, C8.5Q8.4C2, C4.41(C22×Q8), C4⋊C8.329C22, C4⋊C4.272C23, (C2×C8).213C23, (C2×C4).575C24, (C4×C8).125C22, Q8⋊Q8.12C2, C4⋊Q8.204C22, C2.D8.74C22, (C2×Q8).410C23, (C4×Q8).313C22, C4.Q8.117C22, (C2×Q16).173C22, C22.835(C22×D4), C42.C2.73C22, Q8⋊C4.187C22, C2.78(C2×C4○D8), (C2×C4).180(C2×D4), SmallGroup(128,2115)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16⋊6Q8
G = < a,b,c,d | a8=c4=1, b2=a4, d2=c2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 248 in 162 conjugacy classes, 96 normal (24 characteristic)
C1, C2, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C42, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C2×Q8, C4×C8, C4×C8, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2.D8, C4×Q8, C4×Q8, C4×Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C2×Q16, C4×Q16, C4×Q16, C8×Q8, Q8⋊Q8, Q8.Q8, C8.5Q8, C8⋊2Q8, Q8⋊3Q8, Q16⋊6Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4○D8, C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C4○D8, Q8○D8, Q16⋊6Q8
►Regular action on 128 points
Generators in S128
(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)
(1 29 43 107)(2 30 44 108)(3 31 45 109)(4 32 46 110)(5 25 47 111)(6 26 48 112)(7 27 41 105)(8 28 42 106)(9 88 117 33)(10 81 118 34)(11 82 119 35)(12 83 120 36)(13 84 113 37)(14 85 114 38)(15 86 115 39)(16 87 116 40)(17 50 103 68)(18 51 104 69)(19 52 97 70)(20 53 98 71)(21 54 99 72)(22 55 100 65)(23 56 101 66)(24 49 102 67)(57 73 122 93)(58 74 123 94)(59 75 124 95)(60 76 125 96)(61 77 126 89)(62 78 127 90)(63 79 128 91)(64 80 121 92)
(1 76 43 96)(2 77 44 89)(3 78 45 90)(4 79 46 91)(5 80 47 92)(6 73 48 93)(7 74 41 94)(8 75 42 95)(9 99 117 21)(10 100 118 22)(11 101 119 23)(12 102 120 24)(13 103 113 17)(14 104 114 18)(15 97 115 19)(16 98 116 20)(25 64 111 121)(26 57 112 122)(27 58 105 123)(28 59 106 124)(29 60 107 125)(30 61 108 126)(31 62 109 127)(32 63 110 128)(33 72 88 54)(34 65 81 55)(35 66 82 56)(36 67 83 49)(37 68 84 50)(38 69 85 51)(39 70 86 52)(40 71 87 53)
G:=sub<Sym(128)| (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), (1,29,43,107)(2,30,44,108)(3,31,45,109)(4,32,46,110)(5,25,47,111)(6,26,48,112)(7,27,41,105)(8,28,42,106)(9,88,117,33)(10,81,118,34)(11,82,119,35)(12,83,120,36)(13,84,113,37)(14,85,114,38)(15,86,115,39)(16,87,116,40)(17,50,103,68)(18,51,104,69)(19,52,97,70)(20,53,98,71)(21,54,99,72)(22,55,100,65)(23,56,101,66)(24,49,102,67)(57,73,122,93)(58,74,123,94)(59,75,124,95)(60,76,125,96)(61,77,126,89)(62,78,127,90)(63,79,128,91)(64,80,121,92), (1,76,43,96)(2,77,44,89)(3,78,45,90)(4,79,46,91)(5,80,47,92)(6,73,48,93)(7,74,41,94)(8,75,42,95)(9,99,117,21)(10,100,118,22)(11,101,119,23)(12,102,120,24)(13,103,113,17)(14,104,114,18)(15,97,115,19)(16,98,116,20)(25,64,111,121)(26,57,112,122)(27,58,105,123)(28,59,106,124)(29,60,107,125)(30,61,108,126)(31,62,109,127)(32,63,110,128)(33,72,88,54)(34,65,81,55)(35,66,82,56)(36,67,83,49)(37,68,84,50)(38,69,85,51)(39,70,86,52)(40,71,87,53)>;
G:=Group( (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), (1,29,43,107)(2,30,44,108)(3,31,45,109)(4,32,46,110)(5,25,47,111)(6,26,48,112)(7,27,41,105)(8,28,42,106)(9,88,117,33)(10,81,118,34)(11,82,119,35)(12,83,120,36)(13,84,113,37)(14,85,114,38)(15,86,115,39)(16,87,116,40)(17,50,103,68)(18,51,104,69)(19,52,97,70)(20,53,98,71)(21,54,99,72)(22,55,100,65)(23,56,101,66)(24,49,102,67)(57,73,122,93)(58,74,123,94)(59,75,124,95)(60,76,125,96)(61,77,126,89)(62,78,127,90)(63,79,128,91)(64,80,121,92), (1,76,43,96)(2,77,44,89)(3,78,45,90)(4,79,46,91)(5,80,47,92)(6,73,48,93)(7,74,41,94)(8,75,42,95)(9,99,117,21)(10,100,118,22)(11,101,119,23)(12,102,120,24)(13,103,113,17)(14,104,114,18)(15,97,115,19)(16,98,116,20)(25,64,111,121)(26,57,112,122)(27,58,105,123)(28,59,106,124)(29,60,107,125)(30,61,108,126)(31,62,109,127)(32,63,110,128)(33,72,88,54)(34,65,81,55)(35,66,82,56)(36,67,83,49)(37,68,84,50)(38,69,85,51)(39,70,86,52)(40,71,87,53) );
G=PermutationGroup([[(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)], [(1,29,43,107),(2,30,44,108),(3,31,45,109),(4,32,46,110),(5,25,47,111),(6,26,48,112),(7,27,41,105),(8,28,42,106),(9,88,117,33),(10,81,118,34),(11,82,119,35),(12,83,120,36),(13,84,113,37),(14,85,114,38),(15,86,115,39),(16,87,116,40),(17,50,103,68),(18,51,104,69),(19,52,97,70),(20,53,98,71),(21,54,99,72),(22,55,100,65),(23,56,101,66),(24,49,102,67),(57,73,122,93),(58,74,123,94),(59,75,124,95),(60,76,125,96),(61,77,126,89),(62,78,127,90),(63,79,128,91),(64,80,121,92)], [(1,76,43,96),(2,77,44,89),(3,78,45,90),(4,79,46,91),(5,80,47,92),(6,73,48,93),(7,74,41,94),(8,75,42,95),(9,99,117,21),(10,100,118,22),(11,101,119,23),(12,102,120,24),(13,103,113,17),(14,104,114,18),(15,97,115,19),(16,98,116,20),(25,64,111,121),(26,57,112,122),(27,58,105,123),(28,59,106,124),(29,60,107,125),(30,61,108,126),(31,62,109,127),(32,63,110,128),(33,72,88,54),(34,65,81,55),(35,66,82,56),(36,67,83,49),(37,68,84,50),(38,69,85,51),(39,70,86,52),(40,71,87,53)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | C4○D8 | 2- 1+4 | Q8○D8 |
| kernel | Q16⋊6Q8 | C4×Q16 | C8×Q8 | Q8⋊Q8 | Q8.Q8 | C8.5Q8 | C8⋊2Q8 | Q8⋊3Q8 | C4⋊C4 | Q16 | C2×Q8 | C4 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 4 | 2 | 1 | 2 | 3 | 4 | 1 | 8 | 1 | 2 |
Matrix representation of Q16⋊6Q8 ►in GL4(𝔽17) generated by
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 10 | 0 | 0 |
| 10 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 16 | 16 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 7 | 6 |
G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,16,0,0,0,0,16],[1,10,0,0,10,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,1,16,0,0,2,16],[0,4,0,0,13,0,0,0,0,0,11,7,0,0,2,6] >;
Q16⋊6Q8 in GAP, Magma, Sage, TeX
Q_{16}\rtimes_6Q_8 % in TeX
G:=Group("Q16:6Q8"); // GroupNames label
G:=SmallGroup(128,2115);
// by ID
G=gap.SmallGroup(128,2115);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,568,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^4=1,b^2=a^4,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations