direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C42⋊C2, C42⋊4C18, C4⋊C4⋊6C18, (C4×C36)⋊2C2, (C2×C4)⋊4C36, (C2×C36)⋊9C4, (C4×C12).5C6, C36.46(C2×C4), C4.12(C2×C36), C12.53(C2×C12), (C2×C12).21C12, C22⋊C4.3C18, (C22×C4).6C18, C2.3(C22×C36), C22.5(C2×C36), C18.38(C4○D4), (C2×C18).72C23, C23.11(C2×C18), C6.31(C22×C12), (C22×C12).27C6, C18.31(C22×C4), (C22×C36).14C2, (C2×C36).135C22, C22.5(C22×C18), (C22×C18).25C22, (C9×C4⋊C4)⋊15C2, C2.1(C9×C4○D4), (C3×C4⋊C4).24C6, (C2×C4).8(C2×C18), C6.38(C3×C4○D4), C3.(C3×C42⋊C2), (C2×C6).27(C2×C12), (C2×C18).22(C2×C4), (C3×C42⋊C2).C3, (C9×C22⋊C4).6C2, (C2×C12).169(C2×C6), (C3×C22⋊C4).14C6, (C22×C6).43(C2×C6), (C2×C6).77(C22×C6), SmallGroup(288,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C42⋊C2
G = < a,b,c,d | a9=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 138 in 114 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C18, C18, C18, C2×C12, C2×C12, C22×C6, C42⋊C2, C36, C36, C2×C18, C2×C18, C2×C18, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C2×C36, C2×C36, C22×C18, C3×C42⋊C2, C4×C36, C9×C22⋊C4, C9×C4⋊C4, C22×C36, C9×C42⋊C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C9, C12, C2×C6, C22×C4, C4○D4, C18, C2×C12, C22×C6, C42⋊C2, C36, C2×C18, C22×C12, C3×C4○D4, C2×C36, C22×C18, C3×C42⋊C2, C22×C36, C9×C4○D4, C9×C42⋊C2
►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 34 109)(2 107 35 110)(3 108 36 111)(4 100 28 112)(5 101 29 113)(6 102 30 114)(7 103 31 115)(8 104 32 116)(9 105 33 117)(10 79 139 67)(11 80 140 68)(12 81 141 69)(13 73 142 70)(14 74 143 71)(15 75 144 72)(16 76 136 64)(17 77 137 65)(18 78 138 66)(19 84 131 59)(20 85 132 60)(21 86 133 61)(22 87 134 62)(23 88 135 63)(24 89 127 55)(25 90 128 56)(26 82 129 57)(27 83 130 58)(37 124 52 99)(38 125 53 91)(39 126 54 92)(40 118 46 93)(41 119 47 94)(42 120 48 95)(43 121 49 96)(44 122 50 97)(45 123 51 98)
(1 57 40 73)(2 58 41 74)(3 59 42 75)(4 60 43 76)(5 61 44 77)(6 62 45 78)(7 63 37 79)(8 55 38 80)(9 56 39 81)(10 115 135 99)(11 116 127 91)(12 117 128 92)(13 109 129 93)(14 110 130 94)(15 111 131 95)(16 112 132 96)(17 113 133 97)(18 114 134 98)(19 120 144 108)(20 121 136 100)(21 122 137 101)(22 123 138 102)(23 124 139 103)(24 125 140 104)(25 126 141 105)(26 118 142 106)(27 119 143 107)(28 85 49 64)(29 86 50 65)(30 87 51 66)(31 88 52 67)(32 89 53 68)(33 90 54 69)(34 82 46 70)(35 83 47 71)(36 84 48 72)
(10 135)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 144)(20 136)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(91 116)(92 117)(93 109)(94 110)(95 111)(96 112)(97 113)(98 114)(99 115)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 118)(107 119)(108 120)
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), (1,106,34,109)(2,107,35,110)(3,108,36,111)(4,100,28,112)(5,101,29,113)(6,102,30,114)(7,103,31,115)(8,104,32,116)(9,105,33,117)(10,79,139,67)(11,80,140,68)(12,81,141,69)(13,73,142,70)(14,74,143,71)(15,75,144,72)(16,76,136,64)(17,77,137,65)(18,78,138,66)(19,84,131,59)(20,85,132,60)(21,86,133,61)(22,87,134,62)(23,88,135,63)(24,89,127,55)(25,90,128,56)(26,82,129,57)(27,83,130,58)(37,124,52,99)(38,125,53,91)(39,126,54,92)(40,118,46,93)(41,119,47,94)(42,120,48,95)(43,121,49,96)(44,122,50,97)(45,123,51,98), (1,57,40,73)(2,58,41,74)(3,59,42,75)(4,60,43,76)(5,61,44,77)(6,62,45,78)(7,63,37,79)(8,55,38,80)(9,56,39,81)(10,115,135,99)(11,116,127,91)(12,117,128,92)(13,109,129,93)(14,110,130,94)(15,111,131,95)(16,112,132,96)(17,113,133,97)(18,114,134,98)(19,120,144,108)(20,121,136,100)(21,122,137,101)(22,123,138,102)(23,124,139,103)(24,125,140,104)(25,126,141,105)(26,118,142,106)(27,119,143,107)(28,85,49,64)(29,86,50,65)(30,87,51,66)(31,88,52,67)(32,89,53,68)(33,90,54,69)(34,82,46,70)(35,83,47,71)(36,84,48,72), (10,135)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,144)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(91,116)(92,117)(93,109)(94,110)(95,111)(96,112)(97,113)(98,114)(99,115)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,118)(107,119)(108,120)>;
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), (1,106,34,109)(2,107,35,110)(3,108,36,111)(4,100,28,112)(5,101,29,113)(6,102,30,114)(7,103,31,115)(8,104,32,116)(9,105,33,117)(10,79,139,67)(11,80,140,68)(12,81,141,69)(13,73,142,70)(14,74,143,71)(15,75,144,72)(16,76,136,64)(17,77,137,65)(18,78,138,66)(19,84,131,59)(20,85,132,60)(21,86,133,61)(22,87,134,62)(23,88,135,63)(24,89,127,55)(25,90,128,56)(26,82,129,57)(27,83,130,58)(37,124,52,99)(38,125,53,91)(39,126,54,92)(40,118,46,93)(41,119,47,94)(42,120,48,95)(43,121,49,96)(44,122,50,97)(45,123,51,98), (1,57,40,73)(2,58,41,74)(3,59,42,75)(4,60,43,76)(5,61,44,77)(6,62,45,78)(7,63,37,79)(8,55,38,80)(9,56,39,81)(10,115,135,99)(11,116,127,91)(12,117,128,92)(13,109,129,93)(14,110,130,94)(15,111,131,95)(16,112,132,96)(17,113,133,97)(18,114,134,98)(19,120,144,108)(20,121,136,100)(21,122,137,101)(22,123,138,102)(23,124,139,103)(24,125,140,104)(25,126,141,105)(26,118,142,106)(27,119,143,107)(28,85,49,64)(29,86,50,65)(30,87,51,66)(31,88,52,67)(32,89,53,68)(33,90,54,69)(34,82,46,70)(35,83,47,71)(36,84,48,72), (10,135)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,144)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(91,116)(92,117)(93,109)(94,110)(95,111)(96,112)(97,113)(98,114)(99,115)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,118)(107,119)(108,120) );
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)], [(1,106,34,109),(2,107,35,110),(3,108,36,111),(4,100,28,112),(5,101,29,113),(6,102,30,114),(7,103,31,115),(8,104,32,116),(9,105,33,117),(10,79,139,67),(11,80,140,68),(12,81,141,69),(13,73,142,70),(14,74,143,71),(15,75,144,72),(16,76,136,64),(17,77,137,65),(18,78,138,66),(19,84,131,59),(20,85,132,60),(21,86,133,61),(22,87,134,62),(23,88,135,63),(24,89,127,55),(25,90,128,56),(26,82,129,57),(27,83,130,58),(37,124,52,99),(38,125,53,91),(39,126,54,92),(40,118,46,93),(41,119,47,94),(42,120,48,95),(43,121,49,96),(44,122,50,97),(45,123,51,98)], [(1,57,40,73),(2,58,41,74),(3,59,42,75),(4,60,43,76),(5,61,44,77),(6,62,45,78),(7,63,37,79),(8,55,38,80),(9,56,39,81),(10,115,135,99),(11,116,127,91),(12,117,128,92),(13,109,129,93),(14,110,130,94),(15,111,131,95),(16,112,132,96),(17,113,133,97),(18,114,134,98),(19,120,144,108),(20,121,136,100),(21,122,137,101),(22,123,138,102),(23,124,139,103),(24,125,140,104),(25,126,141,105),(26,118,142,106),(27,119,143,107),(28,85,49,64),(29,86,50,65),(30,87,51,66),(31,88,52,67),(32,89,53,68),(33,90,54,69),(34,82,46,70),(35,83,47,71),(36,84,48,72)], [(10,135),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,144),(20,136),(21,137),(22,138),(23,139),(24,140),(25,141),(26,142),(27,143),(91,116),(92,117),(93,109),(94,110),(95,111),(96,112),(97,113),(98,114),(99,115),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,118),(107,119),(108,120)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12AB | 18A | ··· | 18R | 18S | ··· | 18AD | 36A | ··· | 36X | 36Y | ··· | 36CF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C9 | C12 | C18 | C18 | C18 | C18 | C36 | C4○D4 | C3×C4○D4 | C9×C4○D4 |
| kernel | C9×C42⋊C2 | C4×C36 | C9×C22⋊C4 | C9×C4⋊C4 | C22×C36 | C3×C42⋊C2 | C2×C36 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C18 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 6 | 16 | 12 | 12 | 12 | 6 | 48 | 4 | 8 | 24 |
Matrix representation of C9×C42⋊C2 ►in GL4(𝔽37) generated by
| 33 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 2 | 1 |
| 0 | 0 | 32 | 35 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 33 | 36 |
G:=sub<GL(4,GF(37))| [33,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,6,0,0,0,0,2,32,0,0,1,35],[1,0,0,0,0,1,0,0,0,0,31,0,0,0,0,31],[36,0,0,0,0,36,0,0,0,0,1,33,0,0,0,36] >;
C9×C42⋊C2 in GAP, Magma, Sage, TeX
C_9\times C_4^2\rtimes C_2
% in TeX
G:=Group("C9xC4^2:C2"); // GroupNames label
G:=SmallGroup(288,167);
// by ID
G=gap.SmallGroup(288,167);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,365,142,360]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations