direct product, metabelian, supersoluble, monomial
Aliases: C9×C3⋊Q16, C36.47D6, Dic6.2C18, C3⋊C8.C18, (C3×C9)⋊7Q16, C3⋊2(C9×Q16), C4.4(S3×C18), C6.10(D4×C9), Q8.3(S3×C9), C12.52(S3×C6), C12.4(C2×C18), (C3×C18).37D4, (C3×Q8).5C18, (Q8×C9).10S3, (C9×Dic6).4C2, (C3×Dic6).2C6, C32.3(C3×Q16), C18.34(C3⋊D4), (C3×C36).46C22, (Q8×C32).19C6, (C3×C3⋊C8).3C6, (C9×C3⋊C8).2C2, (Q8×C3×C9).3C2, C2.7(C9×C3⋊D4), (C3×C3⋊Q16).C3, (C3×C6).58(C3×D4), C6.48(C3×C3⋊D4), C3.4(C3×C3⋊Q16), (C3×C12).30(C2×C6), (C3×Q8).35(C3×S3), SmallGroup(432,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C3⋊Q16
G = < a,b,c,d | a9=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 120 in 66 conjugacy classes, 33 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, C3×C9, C36, C36, C3×Dic3, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C3×C18, C72, Q8×C9, Q8×C9, C3×C3⋊C8, C3×Dic6, Q8×C32, C9×Dic3, C3×C36, C3×C36, C9×Q16, C3×C3⋊Q16, C9×C3⋊C8, C9×Dic6, Q8×C3×C9, C9×C3⋊Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, Q16, C18, C3×S3, C3⋊D4, C3×D4, C2×C18, S3×C6, C3⋊Q16, C3×Q16, S3×C9, D4×C9, C3×C3⋊D4, S3×C18, C9×Q16, C3×C3⋊Q16, C9×C3⋊D4, C9×C3⋊Q16
►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 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)
(1 135 37 139 31 22 52 18)(2 127 38 140 32 23 53 10)(3 128 39 141 33 24 54 11)(4 129 40 142 34 25 46 12)(5 130 41 143 35 26 47 13)(6 131 42 144 36 27 48 14)(7 132 43 136 28 19 49 15)(8 133 44 137 29 20 50 16)(9 134 45 138 30 21 51 17)(55 104 80 91 89 116 68 125)(56 105 81 92 90 117 69 126)(57 106 73 93 82 109 70 118)(58 107 74 94 83 110 71 119)(59 108 75 95 84 111 72 120)(60 100 76 96 85 112 64 121)(61 101 77 97 86 113 65 122)(62 102 78 98 87 114 66 123)(63 103 79 99 88 115 67 124)
(1 67 31 79)(2 68 32 80)(3 69 33 81)(4 70 34 73)(5 71 35 74)(6 72 36 75)(7 64 28 76)(8 65 29 77)(9 66 30 78)(10 125 140 91)(11 126 141 92)(12 118 142 93)(13 119 143 94)(14 120 144 95)(15 121 136 96)(16 122 137 97)(17 123 138 98)(18 124 139 99)(19 100 132 112)(20 101 133 113)(21 102 134 114)(22 103 135 115)(23 104 127 116)(24 105 128 117)(25 106 129 109)(26 107 130 110)(27 108 131 111)(37 88 52 63)(38 89 53 55)(39 90 54 56)(40 82 46 57)(41 83 47 58)(42 84 48 59)(43 85 49 60)(44 86 50 61)(45 87 51 62)
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), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132)(136,142,139)(137,143,140)(138,144,141), (1,135,37,139,31,22,52,18)(2,127,38,140,32,23,53,10)(3,128,39,141,33,24,54,11)(4,129,40,142,34,25,46,12)(5,130,41,143,35,26,47,13)(6,131,42,144,36,27,48,14)(7,132,43,136,28,19,49,15)(8,133,44,137,29,20,50,16)(9,134,45,138,30,21,51,17)(55,104,80,91,89,116,68,125)(56,105,81,92,90,117,69,126)(57,106,73,93,82,109,70,118)(58,107,74,94,83,110,71,119)(59,108,75,95,84,111,72,120)(60,100,76,96,85,112,64,121)(61,101,77,97,86,113,65,122)(62,102,78,98,87,114,66,123)(63,103,79,99,88,115,67,124), (1,67,31,79)(2,68,32,80)(3,69,33,81)(4,70,34,73)(5,71,35,74)(6,72,36,75)(7,64,28,76)(8,65,29,77)(9,66,30,78)(10,125,140,91)(11,126,141,92)(12,118,142,93)(13,119,143,94)(14,120,144,95)(15,121,136,96)(16,122,137,97)(17,123,138,98)(18,124,139,99)(19,100,132,112)(20,101,133,113)(21,102,134,114)(22,103,135,115)(23,104,127,116)(24,105,128,117)(25,106,129,109)(26,107,130,110)(27,108,131,111)(37,88,52,63)(38,89,53,55)(39,90,54,56)(40,82,46,57)(41,83,47,58)(42,84,48,59)(43,85,49,60)(44,86,50,61)(45,87,51,62)>;
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), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132)(136,142,139)(137,143,140)(138,144,141), (1,135,37,139,31,22,52,18)(2,127,38,140,32,23,53,10)(3,128,39,141,33,24,54,11)(4,129,40,142,34,25,46,12)(5,130,41,143,35,26,47,13)(6,131,42,144,36,27,48,14)(7,132,43,136,28,19,49,15)(8,133,44,137,29,20,50,16)(9,134,45,138,30,21,51,17)(55,104,80,91,89,116,68,125)(56,105,81,92,90,117,69,126)(57,106,73,93,82,109,70,118)(58,107,74,94,83,110,71,119)(59,108,75,95,84,111,72,120)(60,100,76,96,85,112,64,121)(61,101,77,97,86,113,65,122)(62,102,78,98,87,114,66,123)(63,103,79,99,88,115,67,124), (1,67,31,79)(2,68,32,80)(3,69,33,81)(4,70,34,73)(5,71,35,74)(6,72,36,75)(7,64,28,76)(8,65,29,77)(9,66,30,78)(10,125,140,91)(11,126,141,92)(12,118,142,93)(13,119,143,94)(14,120,144,95)(15,121,136,96)(16,122,137,97)(17,123,138,98)(18,124,139,99)(19,100,132,112)(20,101,133,113)(21,102,134,114)(22,103,135,115)(23,104,127,116)(24,105,128,117)(25,106,129,109)(26,107,130,110)(27,108,131,111)(37,88,52,63)(38,89,53,55)(39,90,54,56)(40,82,46,57)(41,83,47,58)(42,84,48,59)(43,85,49,60)(44,86,50,61)(45,87,51,62) );
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)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,97,94),(92,98,95),(93,99,96),(100,106,103),(101,107,104),(102,108,105),(109,115,112),(110,116,113),(111,117,114),(118,124,121),(119,125,122),(120,126,123),(127,133,130),(128,134,131),(129,135,132),(136,142,139),(137,143,140),(138,144,141)], [(1,135,37,139,31,22,52,18),(2,127,38,140,32,23,53,10),(3,128,39,141,33,24,54,11),(4,129,40,142,34,25,46,12),(5,130,41,143,35,26,47,13),(6,131,42,144,36,27,48,14),(7,132,43,136,28,19,49,15),(8,133,44,137,29,20,50,16),(9,134,45,138,30,21,51,17),(55,104,80,91,89,116,68,125),(56,105,81,92,90,117,69,126),(57,106,73,93,82,109,70,118),(58,107,74,94,83,110,71,119),(59,108,75,95,84,111,72,120),(60,100,76,96,85,112,64,121),(61,101,77,97,86,113,65,122),(62,102,78,98,87,114,66,123),(63,103,79,99,88,115,67,124)], [(1,67,31,79),(2,68,32,80),(3,69,33,81),(4,70,34,73),(5,71,35,74),(6,72,36,75),(7,64,28,76),(8,65,29,77),(9,66,30,78),(10,125,140,91),(11,126,141,92),(12,118,142,93),(13,119,143,94),(14,120,144,95),(15,121,136,96),(16,122,137,97),(17,123,138,98),(18,124,139,99),(19,100,132,112),(20,101,133,113),(21,102,134,114),(22,103,135,115),(23,104,127,116),(24,105,128,117),(25,106,129,109),(26,107,130,110),(27,108,131,111),(37,88,52,63),(38,89,53,55),(39,90,54,56),(40,82,46,57),(41,83,47,58),(42,84,48,59),(43,85,49,60),(44,86,50,61),(45,87,51,62)]])
108 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | ··· | 12M | 12N | 12O | 18A | ··· | 18F | 18G | ··· | 18L | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36AD | 36AE | ··· | 36AJ | 72A | ··· | 72L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 12 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | S3 | D4 | D6 | Q16 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×Q16 | S3×C9 | D4×C9 | C3×C3⋊D4 | S3×C18 | C9×Q16 | C9×C3⋊D4 | C3⋊Q16 | C3×C3⋊Q16 | C9×C3⋊Q16 |
| kernel | C9×C3⋊Q16 | C9×C3⋊C8 | C9×Dic6 | Q8×C3×C9 | C3×C3⋊Q16 | C3×C3⋊C8 | C3×Dic6 | Q8×C32 | C3⋊Q16 | C3⋊C8 | Dic6 | C3×Q8 | Q8×C9 | C3×C18 | C36 | C3×C9 | C3×Q8 | C18 | C3×C6 | C12 | C32 | Q8 | C6 | C6 | C4 | C3 | C2 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 4 | 6 | 12 | 12 | 1 | 2 | 6 |
Matrix representation of C9×C3⋊Q16 ►in GL4(𝔽73) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 57 | 57 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 41 | 17 |
| 0 | 0 | 17 | 32 |
G:=sub<GL(4,GF(73))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,0,0,0,0,0,57,57,0,0,16,57],[1,0,0,0,0,72,0,0,0,0,41,17,0,0,17,32] >;
C9×C3⋊Q16 in GAP, Magma, Sage, TeX
C_9\times C_3\rtimes Q_{16} % in TeX
G:=Group("C9xC3:Q16"); // GroupNames label
G:=SmallGroup(432,159);
// by ID
G=gap.SmallGroup(432,159);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,504,197,512,142,2355,1186,192,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations