direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×C8.C4, C8.1C36, C72.5C4, C24.8C12, C36.68D4, M4(2).2C18, C4.8(C2×C36), (C2×C8).5C18, C22.(Q8×C9), C4.19(D4×C9), (C2×C18).2Q8, C36.45(C2×C4), (C2×C24).18C6, (C2×C72).15C2, C12.85(C3×D4), C18.14(C4⋊C4), C12.51(C2×C12), (C3×M4(2)).6C6, (C9×M4(2)).4C2, (C2×C36).118C22, C2.5(C9×C4⋊C4), C6.14(C3×C4⋊C4), C3.(C3×C8.C4), (C2×C6).3(C3×Q8), (C3×C8.C4).C3, (C2×C4).17(C2×C18), (C2×C12).138(C2×C6), SmallGroup(288,58)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C8.C4
G = < a,b,c | a9=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×C8.C4
►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 89 59 95 42 78 53 66)(2 90 60 96 43 79 54 67)(3 82 61 97 44 80 46 68)(4 83 62 98 45 81 47 69)(5 84 63 99 37 73 48 70)(6 85 55 91 38 74 49 71)(7 86 56 92 39 75 50 72)(8 87 57 93 40 76 51 64)(9 88 58 94 41 77 52 65)(10 122 144 108 22 111 34 128)(11 123 136 100 23 112 35 129)(12 124 137 101 24 113 36 130)(13 125 138 102 25 114 28 131)(14 126 139 103 26 115 29 132)(15 118 140 104 27 116 30 133)(16 119 141 105 19 117 31 134)(17 120 142 106 20 109 32 135)(18 121 143 107 21 110 33 127)
(1 131 59 114 42 102 53 125)(2 132 60 115 43 103 54 126)(3 133 61 116 44 104 46 118)(4 134 62 117 45 105 47 119)(5 135 63 109 37 106 48 120)(6 127 55 110 38 107 49 121)(7 128 56 111 39 108 50 122)(8 129 57 112 40 100 51 123)(9 130 58 113 41 101 52 124)(10 86 34 92 22 75 144 72)(11 87 35 93 23 76 136 64)(12 88 36 94 24 77 137 65)(13 89 28 95 25 78 138 66)(14 90 29 96 26 79 139 67)(15 82 30 97 27 80 140 68)(16 83 31 98 19 81 141 69)(17 84 32 99 20 73 142 70)(18 85 33 91 21 74 143 71)
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), (1,89,59,95,42,78,53,66)(2,90,60,96,43,79,54,67)(3,82,61,97,44,80,46,68)(4,83,62,98,45,81,47,69)(5,84,63,99,37,73,48,70)(6,85,55,91,38,74,49,71)(7,86,56,92,39,75,50,72)(8,87,57,93,40,76,51,64)(9,88,58,94,41,77,52,65)(10,122,144,108,22,111,34,128)(11,123,136,100,23,112,35,129)(12,124,137,101,24,113,36,130)(13,125,138,102,25,114,28,131)(14,126,139,103,26,115,29,132)(15,118,140,104,27,116,30,133)(16,119,141,105,19,117,31,134)(17,120,142,106,20,109,32,135)(18,121,143,107,21,110,33,127), (1,131,59,114,42,102,53,125)(2,132,60,115,43,103,54,126)(3,133,61,116,44,104,46,118)(4,134,62,117,45,105,47,119)(5,135,63,109,37,106,48,120)(6,127,55,110,38,107,49,121)(7,128,56,111,39,108,50,122)(8,129,57,112,40,100,51,123)(9,130,58,113,41,101,52,124)(10,86,34,92,22,75,144,72)(11,87,35,93,23,76,136,64)(12,88,36,94,24,77,137,65)(13,89,28,95,25,78,138,66)(14,90,29,96,26,79,139,67)(15,82,30,97,27,80,140,68)(16,83,31,98,19,81,141,69)(17,84,32,99,20,73,142,70)(18,85,33,91,21,74,143,71)>;
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), (1,89,59,95,42,78,53,66)(2,90,60,96,43,79,54,67)(3,82,61,97,44,80,46,68)(4,83,62,98,45,81,47,69)(5,84,63,99,37,73,48,70)(6,85,55,91,38,74,49,71)(7,86,56,92,39,75,50,72)(8,87,57,93,40,76,51,64)(9,88,58,94,41,77,52,65)(10,122,144,108,22,111,34,128)(11,123,136,100,23,112,35,129)(12,124,137,101,24,113,36,130)(13,125,138,102,25,114,28,131)(14,126,139,103,26,115,29,132)(15,118,140,104,27,116,30,133)(16,119,141,105,19,117,31,134)(17,120,142,106,20,109,32,135)(18,121,143,107,21,110,33,127), (1,131,59,114,42,102,53,125)(2,132,60,115,43,103,54,126)(3,133,61,116,44,104,46,118)(4,134,62,117,45,105,47,119)(5,135,63,109,37,106,48,120)(6,127,55,110,38,107,49,121)(7,128,56,111,39,108,50,122)(8,129,57,112,40,100,51,123)(9,130,58,113,41,101,52,124)(10,86,34,92,22,75,144,72)(11,87,35,93,23,76,136,64)(12,88,36,94,24,77,137,65)(13,89,28,95,25,78,138,66)(14,90,29,96,26,79,139,67)(15,82,30,97,27,80,140,68)(16,83,31,98,19,81,141,69)(17,84,32,99,20,73,142,70)(18,85,33,91,21,74,143,71) );
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)], [(1,89,59,95,42,78,53,66),(2,90,60,96,43,79,54,67),(3,82,61,97,44,80,46,68),(4,83,62,98,45,81,47,69),(5,84,63,99,37,73,48,70),(6,85,55,91,38,74,49,71),(7,86,56,92,39,75,50,72),(8,87,57,93,40,76,51,64),(9,88,58,94,41,77,52,65),(10,122,144,108,22,111,34,128),(11,123,136,100,23,112,35,129),(12,124,137,101,24,113,36,130),(13,125,138,102,25,114,28,131),(14,126,139,103,26,115,29,132),(15,118,140,104,27,116,30,133),(16,119,141,105,19,117,31,134),(17,120,142,106,20,109,32,135),(18,121,143,107,21,110,33,127)], [(1,131,59,114,42,102,53,125),(2,132,60,115,43,103,54,126),(3,133,61,116,44,104,46,118),(4,134,62,117,45,105,47,119),(5,135,63,109,37,106,48,120),(6,127,55,110,38,107,49,121),(7,128,56,111,39,108,50,122),(8,129,57,112,40,100,51,123),(9,130,58,113,41,101,52,124),(10,86,34,92,22,75,144,72),(11,87,35,93,23,76,136,64),(12,88,36,94,24,77,137,65),(13,89,28,95,25,78,138,66),(14,90,29,96,26,79,139,67),(15,82,30,97,27,80,140,68),(16,83,31,98,19,81,141,69),(17,84,32,99,20,73,142,70),(18,85,33,91,21,74,143,71)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | ··· | 18F | 18G | ··· | 18L | 24A | ··· | 24H | 24I | ··· | 24P | 36A | ··· | 36L | 36M | ··· | 36R | 72A | ··· | 72X | 72Y | ··· | 72AV |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 | D4 | Q8 | C3×D4 | C3×Q8 | C8.C4 | D4×C9 | Q8×C9 | C3×C8.C4 | C9×C8.C4 |
| kernel | C9×C8.C4 | C2×C72 | C9×M4(2) | C3×C8.C4 | C72 | C2×C24 | C3×M4(2) | C8.C4 | C24 | C2×C8 | M4(2) | C8 | C36 | C2×C18 | C12 | C2×C6 | C9 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 6 | 8 | 6 | 12 | 24 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 8 | 24 |
Matrix representation of C9×C8.C4 ►in GL3(𝔽73) generated by
| 55 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 72 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 51 |
| 27 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 27 | 0 |
G:=sub<GL(3,GF(73))| [55,0,0,0,1,0,0,0,1],[72,0,0,0,63,0,0,0,51],[27,0,0,0,0,27,0,1,0] >;
C9×C8.C4 in GAP, Magma, Sage, TeX
C_9\times C_8.C_4
% in TeX
G:=Group("C9xC8.C4"); // GroupNames label
G:=SmallGroup(288,58);
// by ID
G=gap.SmallGroup(288,58);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,92,268,4371,360,242]);
// Polycyclic
G:=Group<a,b,c|a^9=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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