metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C36.1C8, C72.4C4, C9⋊2M5(2), C8.21D18, C24.87D6, C8.2Dic9, C24.7Dic3, C72.22C22, C4.(C9⋊C8), C9⋊C16⋊5C2, C22.(C9⋊C8), (C2×C8).7D9, C12.2(C3⋊C8), (C2×C18).3C8, C18.9(C2×C8), (C2×C24).24S3, C36.41(C2×C4), (C2×C36).10C4, (C2×C72).10C2, (C2×C4).5Dic9, C3.(C12.C8), C4.10(C2×Dic9), C12.50(C2×Dic3), (C2×C12).17Dic3, C6.9(C2×C3⋊C8), C2.4(C2×C9⋊C8), (C2×C6).4(C3⋊C8), SmallGroup(288,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.C8
G = < a,b | a72=1, b4=a18, bab-1=a53 >
Smallest permutation representation of C36.C8
►On 144 points
Generators in S144
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 123 10 96 19 141 28 114 37 87 46 132 55 105 64 78)(2 104 11 77 20 122 29 95 38 140 47 113 56 86 65 131)(3 85 12 130 21 103 30 76 39 121 48 94 57 139 66 112)(4 138 13 111 22 84 31 129 40 102 49 75 58 120 67 93)(5 119 14 92 23 137 32 110 41 83 50 128 59 101 68 74)(6 100 15 73 24 118 33 91 42 136 51 109 60 82 69 127)(7 81 16 126 25 99 34 144 43 117 52 90 61 135 70 108)(8 134 17 107 26 80 35 125 44 98 53 143 62 116 71 89)(9 115 18 88 27 133 36 106 45 79 54 124 63 97 72 142)
G:=sub<Sym(144)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,123,10,96,19,141,28,114,37,87,46,132,55,105,64,78)(2,104,11,77,20,122,29,95,38,140,47,113,56,86,65,131)(3,85,12,130,21,103,30,76,39,121,48,94,57,139,66,112)(4,138,13,111,22,84,31,129,40,102,49,75,58,120,67,93)(5,119,14,92,23,137,32,110,41,83,50,128,59,101,68,74)(6,100,15,73,24,118,33,91,42,136,51,109,60,82,69,127)(7,81,16,126,25,99,34,144,43,117,52,90,61,135,70,108)(8,134,17,107,26,80,35,125,44,98,53,143,62,116,71,89)(9,115,18,88,27,133,36,106,45,79,54,124,63,97,72,142)>;
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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,123,10,96,19,141,28,114,37,87,46,132,55,105,64,78)(2,104,11,77,20,122,29,95,38,140,47,113,56,86,65,131)(3,85,12,130,21,103,30,76,39,121,48,94,57,139,66,112)(4,138,13,111,22,84,31,129,40,102,49,75,58,120,67,93)(5,119,14,92,23,137,32,110,41,83,50,128,59,101,68,74)(6,100,15,73,24,118,33,91,42,136,51,109,60,82,69,127)(7,81,16,126,25,99,34,144,43,117,52,90,61,135,70,108)(8,134,17,107,26,80,35,125,44,98,53,143,62,116,71,89)(9,115,18,88,27,133,36,106,45,79,54,124,63,97,72,142) );
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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,123,10,96,19,141,28,114,37,87,46,132,55,105,64,78),(2,104,11,77,20,122,29,95,38,140,47,113,56,86,65,131),(3,85,12,130,21,103,30,76,39,121,48,94,57,139,66,112),(4,138,13,111,22,84,31,129,40,102,49,75,58,120,67,93),(5,119,14,92,23,137,32,110,41,83,50,128,59,101,68,74),(6,100,15,73,24,118,33,91,42,136,51,109,60,82,69,127),(7,81,16,126,25,99,34,144,43,117,52,90,61,135,70,108),(8,134,17,107,26,80,35,125,44,98,53,143,62,116,71,89),(9,115,18,88,27,133,36,106,45,79,54,124,63,97,72,142)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | Dic3 | D6 | Dic3 | D9 | C3⋊C8 | C3⋊C8 | M5(2) | Dic9 | D18 | Dic9 | C9⋊C8 | C9⋊C8 | C12.C8 | C36.C8 |
| kernel | C36.C8 | C9⋊C16 | C2×C72 | C72 | C2×C36 | C36 | C2×C18 | C2×C24 | C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C9 | C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 4 | 3 | 3 | 3 | 6 | 6 | 8 | 24 |
Matrix representation of C36.C8 ►in GL2(𝔽433) generated by
| 358 | 0 |
| 0 | 217 |
| 0 | 1 |
| 285 | 0 |
G:=sub<GL(2,GF(433))| [358,0,0,217],[0,285,1,0] >;
C36.C8 in GAP, Magma, Sage, TeX
C_{36}.C_8 % in TeX
G:=Group("C36.C8"); // GroupNames label
G:=SmallGroup(288,19);
// by ID
G=gap.SmallGroup(288,19);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,58,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b|a^72=1,b^4=a^18,b*a*b^-1=a^53>;
// generators/relations
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