p-group, metacyclic, nilpotent (class 3), monomial
Aliases: C8⋊1C16, C8.36D8, C82.2C2, C8.19Q16, C4.5M5(2), (C2×C8).8C8, C4⋊C16.8C2, C4.7(C2×C16), (C4×C8).24C4, C2.4(C4⋊C16), (C2×C8).35Q8, (C2×C8).397D4, C2.2(C8⋊1C8), C4.20(C2.D8), C2.2(C8.C8), C22.17(C4⋊C8), C42.306(C2×C4), (C4×C8).357C22, (C2×C4).57M4(2), C4.13(C8.C4), (C2×C4).72(C2×C8), (C2×C4).157(C4⋊C4), SmallGroup(128,102)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.36D8
G = < a,b,c | a8=b8=1, c2=a, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C8.36D8
►Regular action on 128 points
Generators in S128
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)(65 67 69 71 73 75 77 79)(66 68 70 72 74 76 78 80)(81 83 85 87 89 91 93 95)(82 84 86 88 90 92 94 96)(97 99 101 103 105 107 109 111)(98 100 102 104 106 108 110 112)(113 115 117 119 121 123 125 127)(114 116 118 120 122 124 126 128)
(1 58 85 18 124 102 46 76)(2 77 47 103 125 19 86 59)(3 60 87 20 126 104 48 78)(4 79 33 105 127 21 88 61)(5 62 89 22 128 106 34 80)(6 65 35 107 113 23 90 63)(7 64 91 24 114 108 36 66)(8 67 37 109 115 25 92 49)(9 50 93 26 116 110 38 68)(10 69 39 111 117 27 94 51)(11 52 95 28 118 112 40 70)(12 71 41 97 119 29 96 53)(13 54 81 30 120 98 42 72)(14 73 43 99 121 31 82 55)(15 56 83 32 122 100 44 74)(16 75 45 101 123 17 84 57)
(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)
G:=sub<Sym(128)| (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64)(65,67,69,71,73,75,77,79)(66,68,70,72,74,76,78,80)(81,83,85,87,89,91,93,95)(82,84,86,88,90,92,94,96)(97,99,101,103,105,107,109,111)(98,100,102,104,106,108,110,112)(113,115,117,119,121,123,125,127)(114,116,118,120,122,124,126,128), (1,58,85,18,124,102,46,76)(2,77,47,103,125,19,86,59)(3,60,87,20,126,104,48,78)(4,79,33,105,127,21,88,61)(5,62,89,22,128,106,34,80)(6,65,35,107,113,23,90,63)(7,64,91,24,114,108,36,66)(8,67,37,109,115,25,92,49)(9,50,93,26,116,110,38,68)(10,69,39,111,117,27,94,51)(11,52,95,28,118,112,40,70)(12,71,41,97,119,29,96,53)(13,54,81,30,120,98,42,72)(14,73,43,99,121,31,82,55)(15,56,83,32,122,100,44,74)(16,75,45,101,123,17,84,57), (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)>;
G:=Group( (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64)(65,67,69,71,73,75,77,79)(66,68,70,72,74,76,78,80)(81,83,85,87,89,91,93,95)(82,84,86,88,90,92,94,96)(97,99,101,103,105,107,109,111)(98,100,102,104,106,108,110,112)(113,115,117,119,121,123,125,127)(114,116,118,120,122,124,126,128), (1,58,85,18,124,102,46,76)(2,77,47,103,125,19,86,59)(3,60,87,20,126,104,48,78)(4,79,33,105,127,21,88,61)(5,62,89,22,128,106,34,80)(6,65,35,107,113,23,90,63)(7,64,91,24,114,108,36,66)(8,67,37,109,115,25,92,49)(9,50,93,26,116,110,38,68)(10,69,39,111,117,27,94,51)(11,52,95,28,118,112,40,70)(12,71,41,97,119,29,96,53)(13,54,81,30,120,98,42,72)(14,73,43,99,121,31,82,55)(15,56,83,32,122,100,44,74)(16,75,45,101,123,17,84,57), (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) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16),(17,19,21,23,25,27,29,31),(18,20,22,24,26,28,30,32),(33,35,37,39,41,43,45,47),(34,36,38,40,42,44,46,48),(49,51,53,55,57,59,61,63),(50,52,54,56,58,60,62,64),(65,67,69,71,73,75,77,79),(66,68,70,72,74,76,78,80),(81,83,85,87,89,91,93,95),(82,84,86,88,90,92,94,96),(97,99,101,103,105,107,109,111),(98,100,102,104,106,108,110,112),(113,115,117,119,121,123,125,127),(114,116,118,120,122,124,126,128)], [(1,58,85,18,124,102,46,76),(2,77,47,103,125,19,86,59),(3,60,87,20,126,104,48,78),(4,79,33,105,127,21,88,61),(5,62,89,22,128,106,34,80),(6,65,35,107,113,23,90,63),(7,64,91,24,114,108,36,66),(8,67,37,109,115,25,92,49),(9,50,93,26,116,110,38,68),(10,69,39,111,117,27,94,51),(11,52,95,28,118,112,40,70),(12,71,41,97,119,29,96,53),(13,54,81,30,120,98,42,72),(14,73,43,99,121,31,82,55),(15,56,83,32,122,100,44,74),(16,75,45,101,123,17,84,57)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 8I | ··· | 8AB | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C8 | C16 | D4 | Q8 | D8 | Q16 | M4(2) | C8.C4 | M5(2) | C8.C8 |
| kernel | C8.36D8 | C82 | C4⋊C16 | C4×C8 | C2×C8 | C8 | C2×C8 | C2×C8 | C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C8.36D8 ►in GL4(𝔽17) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 10 | 11 |
| 0 | 0 | 4 | 13 |
| 3 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 5 | 16 |
G:=sub<GL(4,GF(17))| [9,0,0,0,0,9,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,10,4,0,0,11,13],[3,0,0,0,0,14,0,0,0,0,1,5,0,0,4,16] >;
C8.36D8 in GAP, Magma, Sage, TeX
C_8._{36}D_8 % in TeX
G:=Group("C8.36D8"); // GroupNames label
G:=SmallGroup(128,102);
// by ID
G=gap.SmallGroup(128,102);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,148,422,100,136,124]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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