p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊5C8, C42.636C23, C8.4(C2×C8), C2.8(C8×D4), Q8⋊C8.1C2, (C8×Q8).3C2, Q8.2(C2×C8), C8⋊C8.1C2, C8⋊2C8.5C2, (C2×C8).305D4, C2.D8.18C4, C4.11(C8○D4), C4.11(C22×C8), (C4×C8).12C22, (C2×Q16).11C4, (C4×Q16).15C2, C22.81(C4×D4), C2.2(C8.26D4), C4⋊C8.274C22, Q8⋊C4.10C4, C2.1(Q16⋊C4), (C4×Q8).258C22, C4.136(C8.C22), C4⋊C4.130(C2×C4), (C2×C8).124(C2×C4), (C2×C4).1472(C2×D4), (C2×Q8).134(C2×C4), (C2×C4).497(C4○D4), (C2×C4).328(C22×C4), SmallGroup(128,311)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16⋊5C8
G = < a,b,c | a8=c8=1, b2=a4, bab-1=a-1, cac-1=a5, bc=cb >
Subgroups: 120 in 80 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C4, C4, C4, C22, C8, C8, C2×C4, C2×C4, Q8, Q8, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, Q16, C2×Q8, C4×C8, C4×C8, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×Q8, C2×Q16, C8⋊C8, Q8⋊C8, C8⋊2C8, C4×Q16, C8×Q8, Q16⋊5C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8.C22, C8×D4, Q16⋊C4, C8.26D4, Q16⋊5C8
►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 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 37 13 33)(10 36 14 40)(11 35 15 39)(12 34 16 38)(17 62 21 58)(18 61 22 57)(19 60 23 64)(20 59 24 63)(41 67 45 71)(42 66 46 70)(43 65 47 69)(44 72 48 68)(49 74 53 78)(50 73 54 77)(51 80 55 76)(52 79 56 75)(81 109 85 105)(82 108 86 112)(83 107 87 111)(84 106 88 110)(89 118 93 114)(90 117 94 113)(91 116 95 120)(92 115 96 119)(97 122 101 126)(98 121 102 125)(99 128 103 124)(100 127 104 123)
(1 63 55 72 128 114 10 105)(2 60 56 69 121 119 11 110)(3 57 49 66 122 116 12 107)(4 62 50 71 123 113 13 112)(5 59 51 68 124 118 14 109)(6 64 52 65 125 115 15 106)(7 61 53 70 126 120 16 111)(8 58 54 67 127 117 9 108)(17 77 45 104 94 37 86 31)(18 74 46 101 95 34 87 28)(19 79 47 98 96 39 88 25)(20 76 48 103 89 36 81 30)(21 73 41 100 90 33 82 27)(22 78 42 97 91 38 83 32)(23 75 43 102 92 35 84 29)(24 80 44 99 93 40 85 26)
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), (1,30,5,26)(2,29,6,25)(3,28,7,32)(4,27,8,31)(9,37,13,33)(10,36,14,40)(11,35,15,39)(12,34,16,38)(17,62,21,58)(18,61,22,57)(19,60,23,64)(20,59,24,63)(41,67,45,71)(42,66,46,70)(43,65,47,69)(44,72,48,68)(49,74,53,78)(50,73,54,77)(51,80,55,76)(52,79,56,75)(81,109,85,105)(82,108,86,112)(83,107,87,111)(84,106,88,110)(89,118,93,114)(90,117,94,113)(91,116,95,120)(92,115,96,119)(97,122,101,126)(98,121,102,125)(99,128,103,124)(100,127,104,123), (1,63,55,72,128,114,10,105)(2,60,56,69,121,119,11,110)(3,57,49,66,122,116,12,107)(4,62,50,71,123,113,13,112)(5,59,51,68,124,118,14,109)(6,64,52,65,125,115,15,106)(7,61,53,70,126,120,16,111)(8,58,54,67,127,117,9,108)(17,77,45,104,94,37,86,31)(18,74,46,101,95,34,87,28)(19,79,47,98,96,39,88,25)(20,76,48,103,89,36,81,30)(21,73,41,100,90,33,82,27)(22,78,42,97,91,38,83,32)(23,75,43,102,92,35,84,29)(24,80,44,99,93,40,85,26)>;
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), (1,30,5,26)(2,29,6,25)(3,28,7,32)(4,27,8,31)(9,37,13,33)(10,36,14,40)(11,35,15,39)(12,34,16,38)(17,62,21,58)(18,61,22,57)(19,60,23,64)(20,59,24,63)(41,67,45,71)(42,66,46,70)(43,65,47,69)(44,72,48,68)(49,74,53,78)(50,73,54,77)(51,80,55,76)(52,79,56,75)(81,109,85,105)(82,108,86,112)(83,107,87,111)(84,106,88,110)(89,118,93,114)(90,117,94,113)(91,116,95,120)(92,115,96,119)(97,122,101,126)(98,121,102,125)(99,128,103,124)(100,127,104,123), (1,63,55,72,128,114,10,105)(2,60,56,69,121,119,11,110)(3,57,49,66,122,116,12,107)(4,62,50,71,123,113,13,112)(5,59,51,68,124,118,14,109)(6,64,52,65,125,115,15,106)(7,61,53,70,126,120,16,111)(8,58,54,67,127,117,9,108)(17,77,45,104,94,37,86,31)(18,74,46,101,95,34,87,28)(19,79,47,98,96,39,88,25)(20,76,48,103,89,36,81,30)(21,73,41,100,90,33,82,27)(22,78,42,97,91,38,83,32)(23,75,43,102,92,35,84,29)(24,80,44,99,93,40,85,26) );
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)], [(1,30,5,26),(2,29,6,25),(3,28,7,32),(4,27,8,31),(9,37,13,33),(10,36,14,40),(11,35,15,39),(12,34,16,38),(17,62,21,58),(18,61,22,57),(19,60,23,64),(20,59,24,63),(41,67,45,71),(42,66,46,70),(43,65,47,69),(44,72,48,68),(49,74,53,78),(50,73,54,77),(51,80,55,76),(52,79,56,75),(81,109,85,105),(82,108,86,112),(83,107,87,111),(84,106,88,110),(89,118,93,114),(90,117,94,113),(91,116,95,120),(92,115,96,119),(97,122,101,126),(98,121,102,125),(99,128,103,124),(100,127,104,123)], [(1,63,55,72,128,114,10,105),(2,60,56,69,121,119,11,110),(3,57,49,66,122,116,12,107),(4,62,50,71,123,113,13,112),(5,59,51,68,124,118,14,109),(6,64,52,65,125,115,15,106),(7,61,53,70,126,120,16,111),(8,58,54,67,127,117,9,108),(17,77,45,104,94,37,86,31),(18,74,46,101,95,34,87,28),(19,79,47,98,96,39,88,25),(20,76,48,103,89,36,81,30),(21,73,41,100,90,33,82,27),(22,78,42,97,91,38,83,32),(23,75,43,102,92,35,84,29),(24,80,44,99,93,40,85,26)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 8A | ··· | 8H | 8I | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 | C8.C22 | C8.26D4 |
| kernel | Q16⋊5C8 | C8⋊C8 | Q8⋊C8 | C8⋊2C8 | C4×Q16 | C8×Q8 | Q8⋊C4 | C2.D8 | C2×Q16 | Q16 | C2×C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 16 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of Q16⋊5C8 ►in GL6(𝔽17)
| 7 | 4 | 0 | 0 | 0 | 0 |
| 13 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 1 | 1 | 3 |
| 0 | 0 | 16 | 13 | 14 | 1 |
| 0 | 0 | 4 | 12 | 4 | 16 |
| 0 | 0 | 5 | 4 | 1 | 4 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 16 | 6 | 12 |
| 0 | 0 | 16 | 15 | 12 | 11 |
| 0 | 0 | 10 | 3 | 2 | 16 |
| 0 | 0 | 3 | 7 | 16 | 15 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
G:=sub<GL(6,GF(17))| [7,13,0,0,0,0,4,10,0,0,0,0,0,0,13,16,4,5,0,0,1,13,12,4,0,0,1,14,4,1,0,0,3,1,16,4],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,2,16,10,3,0,0,16,15,3,7,0,0,6,12,2,16,0,0,12,11,16,15],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0] >;
Q16⋊5C8 in GAP, Magma, Sage, TeX
Q_{16}\rtimes_5C_8 % in TeX
G:=Group("Q16:5C8"); // GroupNames label
G:=SmallGroup(128,311);
// by ID
G=gap.SmallGroup(128,311);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=c^8=1,b^2=a^4,b*a*b^-1=a^-1,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations