direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C4⋊D4, C36⋊9D4, C4⋊2(D4×C9), C4⋊C4⋊2C18, (C2×C18)⋊4D4, (C2×D4)⋊2C18, C2.5(D4×C18), C6.68(C6×D4), C22⋊2(D4×C9), (D4×C18)⋊11C2, C22⋊C4⋊3C18, (C22×C4)⋊6C18, (C6×D4).10C6, C12.71(C3×D4), C18.68(C2×D4), (C22×C36)⋊11C2, C23.2(C2×C18), C18.41(C4○D4), (C2×C18).76C23, (C22×C12).28C6, (C2×C36).122C22, (C22×C18).27C22, C22.11(C22×C18), (C9×C4⋊C4)⋊11C2, C3.(C3×C4⋊D4), (C3×C4⋊D4).C3, C2.4(C9×C4○D4), (C3×C4⋊C4).12C6, (C2×C4).9(C2×C18), C6.41(C3×C4○D4), (C2×C6).13(C3×D4), (C9×C22⋊C4)⋊11C2, (C2×C12).63(C2×C6), (C3×C22⋊C4).6C6, (C22×C6).46(C2×C6), (C2×C6).81(C22×C6), SmallGroup(288,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4⋊D4
G = < a,b,c,d | a9=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 222 in 141 conjugacy classes, 72 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C18, C18, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C4⋊D4, C36, C36, C2×C18, C2×C18, C2×C18, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C2×C36, C2×C36, C2×C36, D4×C9, C22×C18, C22×C18, C3×C4⋊D4, C9×C22⋊C4, C9×C4⋊C4, C22×C36, D4×C18, D4×C18, C9×C4⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C4○D4, C18, C3×D4, C22×C6, C4⋊D4, C2×C18, C6×D4, C3×C4○D4, D4×C9, C22×C18, C3×C4⋊D4, D4×C18, C9×C4○D4, C9×C4⋊D4
►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 93 41 117)(2 94 42 109)(3 95 43 110)(4 96 44 111)(5 97 45 112)(6 98 37 113)(7 99 38 114)(8 91 39 115)(9 92 40 116)(10 79 127 55)(11 80 128 56)(12 81 129 57)(13 73 130 58)(14 74 131 59)(15 75 132 60)(16 76 133 61)(17 77 134 62)(18 78 135 63)(19 83 144 72)(20 84 136 64)(21 85 137 65)(22 86 138 66)(23 87 139 67)(24 88 140 68)(25 89 141 69)(26 90 142 70)(27 82 143 71)(28 120 48 100)(29 121 49 101)(30 122 50 102)(31 123 51 103)(32 124 52 104)(33 125 53 105)(34 126 54 106)(35 118 46 107)(36 119 47 108)
(1 81 34 70)(2 73 35 71)(3 74 36 72)(4 75 28 64)(5 76 29 65)(6 77 30 66)(7 78 31 67)(8 79 32 68)(9 80 33 69)(10 124 140 91)(11 125 141 92)(12 126 142 93)(13 118 143 94)(14 119 144 95)(15 120 136 96)(16 121 137 97)(17 122 138 98)(18 123 139 99)(19 110 131 108)(20 111 132 100)(21 112 133 101)(22 113 134 102)(23 114 135 103)(24 115 127 104)(25 116 128 105)(26 117 129 106)(27 109 130 107)(37 62 50 86)(38 63 51 87)(39 55 52 88)(40 56 53 89)(41 57 54 90)(42 58 46 82)(43 59 47 83)(44 60 48 84)(45 61 49 85)
(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(55 88)(56 89)(57 90)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 73)(72 74)(91 115)(92 116)(93 117)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 118)(108 119)(127 140)(128 141)(129 142)(130 143)(131 144)(132 136)(133 137)(134 138)(135 139)
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), (1,93,41,117)(2,94,42,109)(3,95,43,110)(4,96,44,111)(5,97,45,112)(6,98,37,113)(7,99,38,114)(8,91,39,115)(9,92,40,116)(10,79,127,55)(11,80,128,56)(12,81,129,57)(13,73,130,58)(14,74,131,59)(15,75,132,60)(16,76,133,61)(17,77,134,62)(18,78,135,63)(19,83,144,72)(20,84,136,64)(21,85,137,65)(22,86,138,66)(23,87,139,67)(24,88,140,68)(25,89,141,69)(26,90,142,70)(27,82,143,71)(28,120,48,100)(29,121,49,101)(30,122,50,102)(31,123,51,103)(32,124,52,104)(33,125,53,105)(34,126,54,106)(35,118,46,107)(36,119,47,108), (1,81,34,70)(2,73,35,71)(3,74,36,72)(4,75,28,64)(5,76,29,65)(6,77,30,66)(7,78,31,67)(8,79,32,68)(9,80,33,69)(10,124,140,91)(11,125,141,92)(12,126,142,93)(13,118,143,94)(14,119,144,95)(15,120,136,96)(16,121,137,97)(17,122,138,98)(18,123,139,99)(19,110,131,108)(20,111,132,100)(21,112,133,101)(22,113,134,102)(23,114,135,103)(24,115,127,104)(25,116,128,105)(26,117,129,106)(27,109,130,107)(37,62,50,86)(38,63,51,87)(39,55,52,88)(40,56,53,89)(41,57,54,90)(42,58,46,82)(43,59,47,83)(44,60,48,84)(45,61,49,85), (10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(55,88)(56,89)(57,90)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,73)(72,74)(91,115)(92,116)(93,117)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,118)(108,119)(127,140)(128,141)(129,142)(130,143)(131,144)(132,136)(133,137)(134,138)(135,139)>;
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), (1,93,41,117)(2,94,42,109)(3,95,43,110)(4,96,44,111)(5,97,45,112)(6,98,37,113)(7,99,38,114)(8,91,39,115)(9,92,40,116)(10,79,127,55)(11,80,128,56)(12,81,129,57)(13,73,130,58)(14,74,131,59)(15,75,132,60)(16,76,133,61)(17,77,134,62)(18,78,135,63)(19,83,144,72)(20,84,136,64)(21,85,137,65)(22,86,138,66)(23,87,139,67)(24,88,140,68)(25,89,141,69)(26,90,142,70)(27,82,143,71)(28,120,48,100)(29,121,49,101)(30,122,50,102)(31,123,51,103)(32,124,52,104)(33,125,53,105)(34,126,54,106)(35,118,46,107)(36,119,47,108), (1,81,34,70)(2,73,35,71)(3,74,36,72)(4,75,28,64)(5,76,29,65)(6,77,30,66)(7,78,31,67)(8,79,32,68)(9,80,33,69)(10,124,140,91)(11,125,141,92)(12,126,142,93)(13,118,143,94)(14,119,144,95)(15,120,136,96)(16,121,137,97)(17,122,138,98)(18,123,139,99)(19,110,131,108)(20,111,132,100)(21,112,133,101)(22,113,134,102)(23,114,135,103)(24,115,127,104)(25,116,128,105)(26,117,129,106)(27,109,130,107)(37,62,50,86)(38,63,51,87)(39,55,52,88)(40,56,53,89)(41,57,54,90)(42,58,46,82)(43,59,47,83)(44,60,48,84)(45,61,49,85), (10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(55,88)(56,89)(57,90)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,73)(72,74)(91,115)(92,116)(93,117)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,118)(108,119)(127,140)(128,141)(129,142)(130,143)(131,144)(132,136)(133,137)(134,138)(135,139) );
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)], [(1,93,41,117),(2,94,42,109),(3,95,43,110),(4,96,44,111),(5,97,45,112),(6,98,37,113),(7,99,38,114),(8,91,39,115),(9,92,40,116),(10,79,127,55),(11,80,128,56),(12,81,129,57),(13,73,130,58),(14,74,131,59),(15,75,132,60),(16,76,133,61),(17,77,134,62),(18,78,135,63),(19,83,144,72),(20,84,136,64),(21,85,137,65),(22,86,138,66),(23,87,139,67),(24,88,140,68),(25,89,141,69),(26,90,142,70),(27,82,143,71),(28,120,48,100),(29,121,49,101),(30,122,50,102),(31,123,51,103),(32,124,52,104),(33,125,53,105),(34,126,54,106),(35,118,46,107),(36,119,47,108)], [(1,81,34,70),(2,73,35,71),(3,74,36,72),(4,75,28,64),(5,76,29,65),(6,77,30,66),(7,78,31,67),(8,79,32,68),(9,80,33,69),(10,124,140,91),(11,125,141,92),(12,126,142,93),(13,118,143,94),(14,119,144,95),(15,120,136,96),(16,121,137,97),(17,122,138,98),(18,123,139,99),(19,110,131,108),(20,111,132,100),(21,112,133,101),(22,113,134,102),(23,114,135,103),(24,115,127,104),(25,116,128,105),(26,117,129,106),(27,109,130,107),(37,62,50,86),(38,63,51,87),(39,55,52,88),(40,56,53,89),(41,57,54,90),(42,58,46,82),(43,59,47,83),(44,60,48,84),(45,61,49,85)], [(10,24),(11,25),(12,26),(13,27),(14,19),(15,20),(16,21),(17,22),(18,23),(55,88),(56,89),(57,90),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,73),(72,74),(91,115),(92,116),(93,117),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,118),(108,119),(127,140),(128,141),(129,142),(130,143),(131,144),(132,136),(133,137),(134,138),(135,139)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 9A | ··· | 9F | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 18A | ··· | 18R | 18S | ··· | 18AD | 18AE | ··· | 18AP | 36A | ··· | 36X | 36Y | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×C4○D4 | D4×C9 | D4×C9 | C9×C4○D4 |
| kernel | C9×C4⋊D4 | C9×C22⋊C4 | C9×C4⋊C4 | C22×C36 | D4×C18 | C3×C4⋊D4 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C36 | C2×C18 | C18 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 2 | 2 | 6 | 6 | 12 | 6 | 6 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 |
Matrix representation of C9×C4⋊D4 ►in GL5(𝔽37)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 31 | 31 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 35 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 1 | 0 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 35 | 36 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 36 |
G:=sub<GL(5,GF(37))| [12,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[36,0,0,0,0,0,31,0,0,0,0,31,6,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,1,35,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,36,0],[36,0,0,0,0,0,1,35,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,36] >;
C9×C4⋊D4 in GAP, Magma, Sage, TeX
C_9\times C_4\rtimes D_4
% in TeX
G:=Group("C9xC4:D4"); // GroupNames label
G:=SmallGroup(288,171);
// by ID
G=gap.SmallGroup(288,171);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,176,1094,360]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations