metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C72.1C4, C36.34D4, C4.18D36, C8.1Dic9, C12.53D12, C24.1Dic3, C22.2Dic18, (C2×C8).5D9, (C2×C72).7C2, C18.4(C4⋊C4), (C2×C18).3Q8, C9⋊1(C8.C4), (C2×C24).14S3, C36.35(C2×C4), (C2×C4).72D18, C4.6(C2×Dic9), C3.(C24.C4), (C2×C12).392D6, C2.3(C4⋊Dic9), C6.7(C4⋊Dic3), (C2×C6).12Dic6, C4.Dic9.1C2, (C2×C36).95C22, C12.41(C2×Dic3), SmallGroup(288,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72.C4
G = < a,b,c | a8=1, b18=a4, c2=b9, ab=ba, cac-1=a3, cbc-1=b17 >
Smallest permutation representation of C72.C4
►On 144 points
Generators in S144
(1 50 28 41 19 68 10 59)(2 51 29 42 20 69 11 60)(3 52 30 43 21 70 12 61)(4 53 31 44 22 71 13 62)(5 54 32 45 23 72 14 63)(6 55 33 46 24 37 15 64)(7 56 34 47 25 38 16 65)(8 57 35 48 26 39 17 66)(9 58 36 49 27 40 18 67)(73 139 82 112 91 121 100 130)(74 140 83 113 92 122 101 131)(75 141 84 114 93 123 102 132)(76 142 85 115 94 124 103 133)(77 143 86 116 95 125 104 134)(78 144 87 117 96 126 105 135)(79 109 88 118 97 127 106 136)(80 110 89 119 98 128 107 137)(81 111 90 120 99 129 108 138)
(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)
(1 106 10 79 19 88 28 97)(2 87 11 96 20 105 29 78)(3 104 12 77 21 86 30 95)(4 85 13 94 22 103 31 76)(5 102 14 75 23 84 32 93)(6 83 15 92 24 101 33 74)(7 100 16 73 25 82 34 91)(8 81 17 90 26 99 35 108)(9 98 18 107 27 80 36 89)(37 140 46 113 55 122 64 131)(38 121 47 130 56 139 65 112)(39 138 48 111 57 120 66 129)(40 119 49 128 58 137 67 110)(41 136 50 109 59 118 68 127)(42 117 51 126 60 135 69 144)(43 134 52 143 61 116 70 125)(44 115 53 124 62 133 71 142)(45 132 54 141 63 114 72 123)
G:=sub<Sym(144)| (1,50,28,41,19,68,10,59)(2,51,29,42,20,69,11,60)(3,52,30,43,21,70,12,61)(4,53,31,44,22,71,13,62)(5,54,32,45,23,72,14,63)(6,55,33,46,24,37,15,64)(7,56,34,47,25,38,16,65)(8,57,35,48,26,39,17,66)(9,58,36,49,27,40,18,67)(73,139,82,112,91,121,100,130)(74,140,83,113,92,122,101,131)(75,141,84,114,93,123,102,132)(76,142,85,115,94,124,103,133)(77,143,86,116,95,125,104,134)(78,144,87,117,96,126,105,135)(79,109,88,118,97,127,106,136)(80,110,89,119,98,128,107,137)(81,111,90,120,99,129,108,138), (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), (1,106,10,79,19,88,28,97)(2,87,11,96,20,105,29,78)(3,104,12,77,21,86,30,95)(4,85,13,94,22,103,31,76)(5,102,14,75,23,84,32,93)(6,83,15,92,24,101,33,74)(7,100,16,73,25,82,34,91)(8,81,17,90,26,99,35,108)(9,98,18,107,27,80,36,89)(37,140,46,113,55,122,64,131)(38,121,47,130,56,139,65,112)(39,138,48,111,57,120,66,129)(40,119,49,128,58,137,67,110)(41,136,50,109,59,118,68,127)(42,117,51,126,60,135,69,144)(43,134,52,143,61,116,70,125)(44,115,53,124,62,133,71,142)(45,132,54,141,63,114,72,123)>;
G:=Group( (1,50,28,41,19,68,10,59)(2,51,29,42,20,69,11,60)(3,52,30,43,21,70,12,61)(4,53,31,44,22,71,13,62)(5,54,32,45,23,72,14,63)(6,55,33,46,24,37,15,64)(7,56,34,47,25,38,16,65)(8,57,35,48,26,39,17,66)(9,58,36,49,27,40,18,67)(73,139,82,112,91,121,100,130)(74,140,83,113,92,122,101,131)(75,141,84,114,93,123,102,132)(76,142,85,115,94,124,103,133)(77,143,86,116,95,125,104,134)(78,144,87,117,96,126,105,135)(79,109,88,118,97,127,106,136)(80,110,89,119,98,128,107,137)(81,111,90,120,99,129,108,138), (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), (1,106,10,79,19,88,28,97)(2,87,11,96,20,105,29,78)(3,104,12,77,21,86,30,95)(4,85,13,94,22,103,31,76)(5,102,14,75,23,84,32,93)(6,83,15,92,24,101,33,74)(7,100,16,73,25,82,34,91)(8,81,17,90,26,99,35,108)(9,98,18,107,27,80,36,89)(37,140,46,113,55,122,64,131)(38,121,47,130,56,139,65,112)(39,138,48,111,57,120,66,129)(40,119,49,128,58,137,67,110)(41,136,50,109,59,118,68,127)(42,117,51,126,60,135,69,144)(43,134,52,143,61,116,70,125)(44,115,53,124,62,133,71,142)(45,132,54,141,63,114,72,123) );
G=PermutationGroup([[(1,50,28,41,19,68,10,59),(2,51,29,42,20,69,11,60),(3,52,30,43,21,70,12,61),(4,53,31,44,22,71,13,62),(5,54,32,45,23,72,14,63),(6,55,33,46,24,37,15,64),(7,56,34,47,25,38,16,65),(8,57,35,48,26,39,17,66),(9,58,36,49,27,40,18,67),(73,139,82,112,91,121,100,130),(74,140,83,113,92,122,101,131),(75,141,84,114,93,123,102,132),(76,142,85,115,94,124,103,133),(77,143,86,116,95,125,104,134),(78,144,87,117,96,126,105,135),(79,109,88,118,97,127,106,136),(80,110,89,119,98,128,107,137),(81,111,90,120,99,129,108,138)], [(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)], [(1,106,10,79,19,88,28,97),(2,87,11,96,20,105,29,78),(3,104,12,77,21,86,30,95),(4,85,13,94,22,103,31,76),(5,102,14,75,23,84,32,93),(6,83,15,92,24,101,33,74),(7,100,16,73,25,82,34,91),(8,81,17,90,26,99,35,108),(9,98,18,107,27,80,36,89),(37,140,46,113,55,122,64,131),(38,121,47,130,56,139,65,112),(39,138,48,111,57,120,66,129),(40,119,49,128,58,137,67,110),(41,136,50,109,59,118,68,127),(42,117,51,126,60,135,69,144),(43,134,52,143,61,116,70,125),(44,115,53,124,62,133,71,142),(45,132,54,141,63,114,72,123)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 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 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 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 | S3 | D4 | Q8 | Dic3 | D6 | D9 | D12 | Dic6 | C8.C4 | Dic9 | D18 | D36 | Dic18 | C24.C4 | C72.C4 |
| kernel | C72.C4 | C4.Dic9 | C2×C72 | C72 | C2×C24 | C36 | C2×C18 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C9 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 4 | 6 | 3 | 6 | 6 | 8 | 24 |
Matrix representation of C72.C4 ►in GL2(𝔽73) generated by
| 51 | 0 |
| 0 | 10 |
| 54 | 0 |
| 0 | 50 |
| 0 | 1 |
| 27 | 0 |
G:=sub<GL(2,GF(73))| [51,0,0,10],[54,0,0,50],[0,27,1,0] >;
C72.C4 in GAP, Magma, Sage, TeX
C_{72}.C_4 % in TeX
G:=Group("C72.C4"); // GroupNames label
G:=SmallGroup(288,20);
// by ID
G=gap.SmallGroup(288,20);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,100,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^18=a^4,c^2=b^9,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^17>;
// generators/relations
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