direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×2- 1+4, C18.20C24, C36.53C23, C4○D4⋊6C18, (C2×Q8)⋊7C18, (Q8×C18)⋊12C2, D4.4(C2×C18), (C6×Q8).24C6, Q8.7(C2×C18), C2.5(C23×C18), C6.20(C23×C6), (C2×C18).8C23, C3.(C3×2- 1+4), C4.10(C22×C18), (C2×C36).69C22, C12.54(C22×C6), (D4×C9).14C22, (Q8×C9).15C22, C22.2(C22×C18), (C3×2- 1+4).2C3, (C9×C4○D4)⋊9C2, (C2×C4).6(C2×C18), (C2×C12).70(C2×C6), (C3×C4○D4).18C6, (C3×D4).22(C2×C6), (C3×Q8).35(C2×C6), (C2×C6).10(C22×C6), SmallGroup(288,372)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×2- 1+4
G = < a,b,c,d,e | a9=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 234 in 219 conjugacy classes, 204 normal (9 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, D4, Q8, C9, C12, C2×C6, C2×Q8, C4○D4, C18, C18, C2×C12, C3×D4, C3×Q8, 2- 1+4, C36, C2×C18, C6×Q8, C3×C4○D4, C2×C36, D4×C9, Q8×C9, C3×2- 1+4, Q8×C18, C9×C4○D4, C9×2- 1+4
Quotients: C1, C2, C3, C22, C6, C23, C9, C2×C6, C24, C18, C22×C6, 2- 1+4, C2×C18, C23×C6, C22×C18, C3×2- 1+4, C23×C18, C9×2- 1+4
►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)
(1 106 23 113)(2 107 24 114)(3 108 25 115)(4 100 26 116)(5 101 27 117)(6 102 19 109)(7 103 20 110)(8 104 21 111)(9 105 22 112)(10 48 138 55)(11 49 139 56)(12 50 140 57)(13 51 141 58)(14 52 142 59)(15 53 143 60)(16 54 144 61)(17 46 136 62)(18 47 137 63)(28 98 44 82)(29 99 45 83)(30 91 37 84)(31 92 38 85)(32 93 39 86)(33 94 40 87)(34 95 41 88)(35 96 42 89)(36 97 43 90)(64 134 80 118)(65 135 81 119)(66 127 73 120)(67 128 74 121)(68 129 75 122)(69 130 76 123)(70 131 77 124)(71 132 78 125)(72 133 79 126)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 133)(17 134)(18 135)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 81)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(97 115)(98 116)(99 117)(118 136)(119 137)(120 138)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(1 77 23 70)(2 78 24 71)(3 79 25 72)(4 80 26 64)(5 81 27 65)(6 73 19 66)(7 74 20 67)(8 75 21 68)(9 76 22 69)(10 84 138 91)(11 85 139 92)(12 86 140 93)(13 87 141 94)(14 88 142 95)(15 89 143 96)(16 90 144 97)(17 82 136 98)(18 83 137 99)(28 62 44 46)(29 63 45 47)(30 55 37 48)(31 56 38 49)(32 57 39 50)(33 58 40 51)(34 59 41 52)(35 60 42 53)(36 61 43 54)(100 118 116 134)(101 119 117 135)(102 120 109 127)(103 121 110 128)(104 122 111 129)(105 123 112 130)(106 124 113 131)(107 125 114 132)(108 126 115 133)
(1 131 23 124)(2 132 24 125)(3 133 25 126)(4 134 26 118)(5 135 27 119)(6 127 19 120)(7 128 20 121)(8 129 21 122)(9 130 22 123)(10 37 138 30)(11 38 139 31)(12 39 140 32)(13 40 141 33)(14 41 142 34)(15 42 143 35)(16 43 144 36)(17 44 136 28)(18 45 137 29)(46 82 62 98)(47 83 63 99)(48 84 55 91)(49 85 56 92)(50 86 57 93)(51 87 58 94)(52 88 59 95)(53 89 60 96)(54 90 61 97)(64 100 80 116)(65 101 81 117)(66 102 73 109)(67 103 74 110)(68 104 75 111)(69 105 76 112)(70 106 77 113)(71 107 78 114)(72 108 79 115)
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), (1,106,23,113)(2,107,24,114)(3,108,25,115)(4,100,26,116)(5,101,27,117)(6,102,19,109)(7,103,20,110)(8,104,21,111)(9,105,22,112)(10,48,138,55)(11,49,139,56)(12,50,140,57)(13,51,141,58)(14,52,142,59)(15,53,143,60)(16,54,144,61)(17,46,136,62)(18,47,137,63)(28,98,44,82)(29,99,45,83)(30,91,37,84)(31,92,38,85)(32,93,39,86)(33,94,40,87)(34,95,41,88)(35,96,42,89)(36,97,43,90)(64,134,80,118)(65,135,81,119)(66,127,73,120)(67,128,74,121)(68,129,75,122)(69,130,76,123)(70,131,77,124)(71,132,78,125)(72,133,79,126), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (1,77,23,70)(2,78,24,71)(3,79,25,72)(4,80,26,64)(5,81,27,65)(6,73,19,66)(7,74,20,67)(8,75,21,68)(9,76,22,69)(10,84,138,91)(11,85,139,92)(12,86,140,93)(13,87,141,94)(14,88,142,95)(15,89,143,96)(16,90,144,97)(17,82,136,98)(18,83,137,99)(28,62,44,46)(29,63,45,47)(30,55,37,48)(31,56,38,49)(32,57,39,50)(33,58,40,51)(34,59,41,52)(35,60,42,53)(36,61,43,54)(100,118,116,134)(101,119,117,135)(102,120,109,127)(103,121,110,128)(104,122,111,129)(105,123,112,130)(106,124,113,131)(107,125,114,132)(108,126,115,133), (1,131,23,124)(2,132,24,125)(3,133,25,126)(4,134,26,118)(5,135,27,119)(6,127,19,120)(7,128,20,121)(8,129,21,122)(9,130,22,123)(10,37,138,30)(11,38,139,31)(12,39,140,32)(13,40,141,33)(14,41,142,34)(15,42,143,35)(16,43,144,36)(17,44,136,28)(18,45,137,29)(46,82,62,98)(47,83,63,99)(48,84,55,91)(49,85,56,92)(50,86,57,93)(51,87,58,94)(52,88,59,95)(53,89,60,96)(54,90,61,97)(64,100,80,116)(65,101,81,117)(66,102,73,109)(67,103,74,110)(68,104,75,111)(69,105,76,112)(70,106,77,113)(71,107,78,114)(72,108,79,115)>;
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), (1,106,23,113)(2,107,24,114)(3,108,25,115)(4,100,26,116)(5,101,27,117)(6,102,19,109)(7,103,20,110)(8,104,21,111)(9,105,22,112)(10,48,138,55)(11,49,139,56)(12,50,140,57)(13,51,141,58)(14,52,142,59)(15,53,143,60)(16,54,144,61)(17,46,136,62)(18,47,137,63)(28,98,44,82)(29,99,45,83)(30,91,37,84)(31,92,38,85)(32,93,39,86)(33,94,40,87)(34,95,41,88)(35,96,42,89)(36,97,43,90)(64,134,80,118)(65,135,81,119)(66,127,73,120)(67,128,74,121)(68,129,75,122)(69,130,76,123)(70,131,77,124)(71,132,78,125)(72,133,79,126), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(97,115)(98,116)(99,117)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (1,77,23,70)(2,78,24,71)(3,79,25,72)(4,80,26,64)(5,81,27,65)(6,73,19,66)(7,74,20,67)(8,75,21,68)(9,76,22,69)(10,84,138,91)(11,85,139,92)(12,86,140,93)(13,87,141,94)(14,88,142,95)(15,89,143,96)(16,90,144,97)(17,82,136,98)(18,83,137,99)(28,62,44,46)(29,63,45,47)(30,55,37,48)(31,56,38,49)(32,57,39,50)(33,58,40,51)(34,59,41,52)(35,60,42,53)(36,61,43,54)(100,118,116,134)(101,119,117,135)(102,120,109,127)(103,121,110,128)(104,122,111,129)(105,123,112,130)(106,124,113,131)(107,125,114,132)(108,126,115,133), (1,131,23,124)(2,132,24,125)(3,133,25,126)(4,134,26,118)(5,135,27,119)(6,127,19,120)(7,128,20,121)(8,129,21,122)(9,130,22,123)(10,37,138,30)(11,38,139,31)(12,39,140,32)(13,40,141,33)(14,41,142,34)(15,42,143,35)(16,43,144,36)(17,44,136,28)(18,45,137,29)(46,82,62,98)(47,83,63,99)(48,84,55,91)(49,85,56,92)(50,86,57,93)(51,87,58,94)(52,88,59,95)(53,89,60,96)(54,90,61,97)(64,100,80,116)(65,101,81,117)(66,102,73,109)(67,103,74,110)(68,104,75,111)(69,105,76,112)(70,106,77,113)(71,107,78,114)(72,108,79,115) );
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)], [(1,106,23,113),(2,107,24,114),(3,108,25,115),(4,100,26,116),(5,101,27,117),(6,102,19,109),(7,103,20,110),(8,104,21,111),(9,105,22,112),(10,48,138,55),(11,49,139,56),(12,50,140,57),(13,51,141,58),(14,52,142,59),(15,53,143,60),(16,54,144,61),(17,46,136,62),(18,47,137,63),(28,98,44,82),(29,99,45,83),(30,91,37,84),(31,92,38,85),(32,93,39,86),(33,94,40,87),(34,95,41,88),(35,96,42,89),(36,97,43,90),(64,134,80,118),(65,135,81,119),(66,127,73,120),(67,128,74,121),(68,129,75,122),(69,130,76,123),(70,131,77,124),(71,132,78,125),(72,133,79,126)], [(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,127),(11,128),(12,129),(13,130),(14,131),(15,132),(16,133),(17,134),(18,135),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,79),(62,80),(63,81),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,113),(96,114),(97,115),(98,116),(99,117),(118,136),(119,137),(120,138),(121,139),(122,140),(123,141),(124,142),(125,143),(126,144)], [(1,77,23,70),(2,78,24,71),(3,79,25,72),(4,80,26,64),(5,81,27,65),(6,73,19,66),(7,74,20,67),(8,75,21,68),(9,76,22,69),(10,84,138,91),(11,85,139,92),(12,86,140,93),(13,87,141,94),(14,88,142,95),(15,89,143,96),(16,90,144,97),(17,82,136,98),(18,83,137,99),(28,62,44,46),(29,63,45,47),(30,55,37,48),(31,56,38,49),(32,57,39,50),(33,58,40,51),(34,59,41,52),(35,60,42,53),(36,61,43,54),(100,118,116,134),(101,119,117,135),(102,120,109,127),(103,121,110,128),(104,122,111,129),(105,123,112,130),(106,124,113,131),(107,125,114,132),(108,126,115,133)], [(1,131,23,124),(2,132,24,125),(3,133,25,126),(4,134,26,118),(5,135,27,119),(6,127,19,120),(7,128,20,121),(8,129,21,122),(9,130,22,123),(10,37,138,30),(11,38,139,31),(12,39,140,32),(13,40,141,33),(14,41,142,34),(15,42,143,35),(16,43,144,36),(17,44,136,28),(18,45,137,29),(46,82,62,98),(47,83,63,99),(48,84,55,91),(49,85,56,92),(50,86,57,93),(51,87,58,94),(52,88,59,95),(53,89,60,96),(54,90,61,97),(64,100,80,116),(65,101,81,117),(66,102,73,109),(67,103,74,110),(68,104,75,111),(69,105,76,112),(70,106,77,113),(71,107,78,114),(72,108,79,115)]])
153 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 3A | 3B | 4A | ··· | 4J | 6A | 6B | 6C | ··· | 6L | 9A | ··· | 9F | 12A | ··· | 12T | 18A | ··· | 18F | 18G | ··· | 18AJ | 36A | ··· | 36BH |
| order | 1 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
153 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | 2- 1+4 | C3×2- 1+4 | C9×2- 1+4 |
| kernel | C9×2- 1+4 | Q8×C18 | C9×C4○D4 | C3×2- 1+4 | C6×Q8 | C3×C4○D4 | 2- 1+4 | C2×Q8 | C4○D4 | C9 | C3 | C1 |
| # reps | 1 | 5 | 10 | 2 | 10 | 20 | 6 | 30 | 60 | 1 | 2 | 6 |
Matrix representation of C9×2- 1+4 ►in GL4(𝔽37) generated by
| 33 | 0 | 0 | 0 |
| 0 | 33 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 0 | 33 |
| 27 | 0 | 19 | 0 |
| 3 | 25 | 27 | 22 |
| 20 | 0 | 10 | 0 |
| 2 | 22 | 15 | 12 |
| 33 | 35 | 0 | 0 |
| 26 | 4 | 0 | 0 |
| 0 | 23 | 1 | 14 |
| 26 | 5 | 0 | 36 |
| 33 | 0 | 21 | 35 |
| 26 | 0 | 3 | 4 |
| 18 | 23 | 36 | 23 |
| 31 | 1 | 3 | 5 |
| 12 | 30 | 30 | 13 |
| 33 | 14 | 36 | 11 |
| 26 | 8 | 3 | 5 |
| 32 | 24 | 36 | 8 |
G:=sub<GL(4,GF(37))| [33,0,0,0,0,33,0,0,0,0,33,0,0,0,0,33],[27,3,20,2,0,25,0,22,19,27,10,15,0,22,0,12],[33,26,0,26,35,4,23,5,0,0,1,0,0,0,14,36],[33,26,18,31,0,0,23,1,21,3,36,3,35,4,23,5],[12,33,26,32,30,14,8,24,30,36,3,36,13,11,5,8] >;
C9×2- 1+4 in GAP, Magma, Sage, TeX
C_9\times 2_-^{1+4} % in TeX
G:=Group("C9xES-(2,2)"); // GroupNames label
G:=SmallGroup(288,372);
// by ID
G=gap.SmallGroup(288,372);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,701,344,555,268,1571,242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations