direct product, metacyclic, supersoluble, monomial
Aliases: C9×C24⋊C2, C72⋊6S3, C24⋊2C18, C36.71D6, Dic6⋊1C18, D12.1C18, C18.19D12, C8⋊2(S3×C9), C6.1(D4×C9), (C3×C9)⋊7SD16, (C3×C72)⋊12C2, C4.8(S3×C18), C3⋊1(C9×SD16), C2.3(C9×D12), C24.24(C3×S3), (C3×C24).15C6, C12.8(C2×C18), (C9×Dic6)⋊7C2, (C9×D12).3C2, (C3×D12).8C6, (C3×C18).19D4, C6.31(C3×D12), C12.106(S3×C6), (C3×Dic6).8C6, (C3×C36).73C22, C32.2(C3×SD16), (C3×C24⋊C2).C3, C3.4(C3×C24⋊C2), (C3×C6).39(C3×D4), (C3×C12).77(C2×C6), SmallGroup(432,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C24⋊C2
G = < a,b,c | a9=b24=c2=1, ab=ba, ac=ca, cbc=b11 >
Subgroups: 172 in 68 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, SD16, C18, C18, C3×S3, C3×C6, C24, C24, Dic6, D12, C3×D4, C3×Q8, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, C24⋊C2, C3×SD16, S3×C9, C3×C18, C72, C72, D4×C9, Q8×C9, C3×C24, C3×Dic6, C3×D12, C9×Dic3, C3×C36, S3×C18, C9×SD16, C3×C24⋊C2, C3×C72, C9×Dic6, C9×D12, C9×C24⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, SD16, C18, C3×S3, D12, C3×D4, C2×C18, S3×C6, C24⋊C2, C3×SD16, S3×C9, D4×C9, C3×D12, S3×C18, C9×SD16, C3×C24⋊C2, C9×D12, C9×C24⋊C2
►On 144 points
Generators in S144
(1 103 42 9 111 26 17 119 34)(2 104 43 10 112 27 18 120 35)(3 105 44 11 113 28 19 97 36)(4 106 45 12 114 29 20 98 37)(5 107 46 13 115 30 21 99 38)(6 108 47 14 116 31 22 100 39)(7 109 48 15 117 32 23 101 40)(8 110 25 16 118 33 24 102 41)(49 73 140 65 89 132 57 81 124)(50 74 141 66 90 133 58 82 125)(51 75 142 67 91 134 59 83 126)(52 76 143 68 92 135 60 84 127)(53 77 144 69 93 136 61 85 128)(54 78 121 70 94 137 62 86 129)(55 79 122 71 95 138 63 87 130)(56 80 123 72 96 139 64 88 131)
(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 66)(2 53)(3 64)(4 51)(5 62)(6 49)(7 60)(8 71)(9 58)(10 69)(11 56)(12 67)(13 54)(14 65)(15 52)(16 63)(17 50)(18 61)(19 72)(20 59)(21 70)(22 57)(23 68)(24 55)(25 138)(26 125)(27 136)(28 123)(29 134)(30 121)(31 132)(32 143)(33 130)(34 141)(35 128)(36 139)(37 126)(38 137)(39 124)(40 135)(41 122)(42 133)(43 144)(44 131)(45 142)(46 129)(47 140)(48 127)(73 108)(74 119)(75 106)(76 117)(77 104)(78 115)(79 102)(80 113)(81 100)(82 111)(83 98)(84 109)(85 120)(86 107)(87 118)(88 105)(89 116)(90 103)(91 114)(92 101)(93 112)(94 99)(95 110)(96 97)
G:=sub<Sym(144)| (1,103,42,9,111,26,17,119,34)(2,104,43,10,112,27,18,120,35)(3,105,44,11,113,28,19,97,36)(4,106,45,12,114,29,20,98,37)(5,107,46,13,115,30,21,99,38)(6,108,47,14,116,31,22,100,39)(7,109,48,15,117,32,23,101,40)(8,110,25,16,118,33,24,102,41)(49,73,140,65,89,132,57,81,124)(50,74,141,66,90,133,58,82,125)(51,75,142,67,91,134,59,83,126)(52,76,143,68,92,135,60,84,127)(53,77,144,69,93,136,61,85,128)(54,78,121,70,94,137,62,86,129)(55,79,122,71,95,138,63,87,130)(56,80,123,72,96,139,64,88,131), (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), (1,66)(2,53)(3,64)(4,51)(5,62)(6,49)(7,60)(8,71)(9,58)(10,69)(11,56)(12,67)(13,54)(14,65)(15,52)(16,63)(17,50)(18,61)(19,72)(20,59)(21,70)(22,57)(23,68)(24,55)(25,138)(26,125)(27,136)(28,123)(29,134)(30,121)(31,132)(32,143)(33,130)(34,141)(35,128)(36,139)(37,126)(38,137)(39,124)(40,135)(41,122)(42,133)(43,144)(44,131)(45,142)(46,129)(47,140)(48,127)(73,108)(74,119)(75,106)(76,117)(77,104)(78,115)(79,102)(80,113)(81,100)(82,111)(83,98)(84,109)(85,120)(86,107)(87,118)(88,105)(89,116)(90,103)(91,114)(92,101)(93,112)(94,99)(95,110)(96,97)>;
G:=Group( (1,103,42,9,111,26,17,119,34)(2,104,43,10,112,27,18,120,35)(3,105,44,11,113,28,19,97,36)(4,106,45,12,114,29,20,98,37)(5,107,46,13,115,30,21,99,38)(6,108,47,14,116,31,22,100,39)(7,109,48,15,117,32,23,101,40)(8,110,25,16,118,33,24,102,41)(49,73,140,65,89,132,57,81,124)(50,74,141,66,90,133,58,82,125)(51,75,142,67,91,134,59,83,126)(52,76,143,68,92,135,60,84,127)(53,77,144,69,93,136,61,85,128)(54,78,121,70,94,137,62,86,129)(55,79,122,71,95,138,63,87,130)(56,80,123,72,96,139,64,88,131), (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), (1,66)(2,53)(3,64)(4,51)(5,62)(6,49)(7,60)(8,71)(9,58)(10,69)(11,56)(12,67)(13,54)(14,65)(15,52)(16,63)(17,50)(18,61)(19,72)(20,59)(21,70)(22,57)(23,68)(24,55)(25,138)(26,125)(27,136)(28,123)(29,134)(30,121)(31,132)(32,143)(33,130)(34,141)(35,128)(36,139)(37,126)(38,137)(39,124)(40,135)(41,122)(42,133)(43,144)(44,131)(45,142)(46,129)(47,140)(48,127)(73,108)(74,119)(75,106)(76,117)(77,104)(78,115)(79,102)(80,113)(81,100)(82,111)(83,98)(84,109)(85,120)(86,107)(87,118)(88,105)(89,116)(90,103)(91,114)(92,101)(93,112)(94,99)(95,110)(96,97) );
G=PermutationGroup([[(1,103,42,9,111,26,17,119,34),(2,104,43,10,112,27,18,120,35),(3,105,44,11,113,28,19,97,36),(4,106,45,12,114,29,20,98,37),(5,107,46,13,115,30,21,99,38),(6,108,47,14,116,31,22,100,39),(7,109,48,15,117,32,23,101,40),(8,110,25,16,118,33,24,102,41),(49,73,140,65,89,132,57,81,124),(50,74,141,66,90,133,58,82,125),(51,75,142,67,91,134,59,83,126),(52,76,143,68,92,135,60,84,127),(53,77,144,69,93,136,61,85,128),(54,78,121,70,94,137,62,86,129),(55,79,122,71,95,138,63,87,130),(56,80,123,72,96,139,64,88,131)], [(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)], [(1,66),(2,53),(3,64),(4,51),(5,62),(6,49),(7,60),(8,71),(9,58),(10,69),(11,56),(12,67),(13,54),(14,65),(15,52),(16,63),(17,50),(18,61),(19,72),(20,59),(21,70),(22,57),(23,68),(24,55),(25,138),(26,125),(27,136),(28,123),(29,134),(30,121),(31,132),(32,143),(33,130),(34,141),(35,128),(36,139),(37,126),(38,137),(39,124),(40,135),(41,122),(42,133),(43,144),(44,131),(45,142),(46,129),(47,140),(48,127),(73,108),(74,119),(75,106),(76,117),(77,104),(78,115),(79,102),(80,113),(81,100),(82,111),(83,98),(84,109),(85,120),(86,107),(87,118),(88,105),(89,116),(90,103),(91,114),(92,101),(93,112),(94,99),(95,110),(96,97)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12H | 12I | 12J | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18R | 24A | ··· | 24P | 36A | ··· | 36R | 36S | ··· | 36X | 72A | ··· | 72AJ |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
135 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 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | S3 | D4 | D6 | SD16 | C3×S3 | D12 | C3×D4 | S3×C6 | C24⋊C2 | C3×SD16 | S3×C9 | D4×C9 | C3×D12 | S3×C18 | C9×SD16 | C3×C24⋊C2 | C9×D12 | C9×C24⋊C2 |
| kernel | C9×C24⋊C2 | C3×C72 | C9×Dic6 | C9×D12 | C3×C24⋊C2 | C3×C24 | C3×Dic6 | C3×D12 | C24⋊C2 | C24 | Dic6 | D12 | C72 | C3×C18 | C36 | C3×C9 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C9×C24⋊C2 ►in GL4(𝔽73) generated by
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 3 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 6 | 67 |
| 0 | 0 | 6 | 6 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 |
| 0 | 0 | 16 | 57 |
G:=sub<GL(4,GF(73))| [32,0,0,0,0,32,0,0,0,0,64,0,0,0,0,64],[3,0,0,0,0,49,0,0,0,0,6,6,0,0,67,6],[0,1,0,0,1,0,0,0,0,0,16,16,0,0,16,57] >;
C9×C24⋊C2 in GAP, Magma, Sage, TeX
C_9\times C_{24}\rtimes C_2 % in TeX
G:=Group("C9xC24:C2"); // GroupNames label
G:=SmallGroup(432,111);
// by ID
G=gap.SmallGroup(432,111);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,92,142,2355,192,14118]);
// Polycyclic
G:=Group<a,b,c|a^9=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations