direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×C4○D8, D8⋊3C18, Q16⋊3C18, C36.69D4, SD16⋊3C18, C72.28C22, C36.47C23, (C2×C8)⋊4C18, (C9×D8)⋊7C2, (C2×C72)⋊12C2, C4○D4⋊3C18, C8.6(C2×C18), (C9×Q16)⋊7C2, (C3×D8).7C6, C4.20(D4×C9), C6.77(C6×D4), (C2×C24).19C6, C24.26(C2×C6), (C9×SD16)⋊7C2, D4.2(C2×C18), C12.87(C3×D4), (C2×C18).11D4, C18.77(C2×D4), C2.14(D4×C18), Q8.5(C2×C18), (C3×Q16).7C6, C22.1(D4×C9), C4.4(C22×C18), (C3×SD16).4C6, C12.47(C22×C6), (D4×C9).12C22, (Q8×C9).13C22, (C2×C36).130C22, C3.(C3×C4○D8), (C3×C4○D8).C3, (C9×C4○D4)⋊6C2, (C2×C6).14(C3×D4), (C2×C4).29(C2×C18), (C3×C4○D4).13C6, (C3×D4).13(C2×C6), (C3×Q8).26(C2×C6), (C2×C12).147(C2×C6), SmallGroup(288,185)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4○D8
G = < a,b,c,d | a9=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
Subgroups: 138 in 93 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C9, C12, C12, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C18, C18, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C36, C36, C2×C18, C2×C18, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C72, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, C3×C4○D8, C2×C72, C9×D8, C9×SD16, C9×Q16, C9×C4○D4, C9×C4○D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C18, C3×D4, C22×C6, C4○D8, C2×C18, C6×D4, D4×C9, C22×C18, C3×C4○D8, D4×C18, C9×C4○D8
►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 78 42 89)(2 79 43 90)(3 80 44 82)(4 81 45 83)(5 73 37 84)(6 74 38 85)(7 75 39 86)(8 76 40 87)(9 77 41 88)(10 122 20 111)(11 123 21 112)(12 124 22 113)(13 125 23 114)(14 126 24 115)(15 118 25 116)(16 119 26 117)(17 120 27 109)(18 121 19 110)(28 131 138 102)(29 132 139 103)(30 133 140 104)(31 134 141 105)(32 135 142 106)(33 127 143 107)(34 128 144 108)(35 129 136 100)(36 130 137 101)(46 95 59 66)(47 96 60 67)(48 97 61 68)(49 98 62 69)(50 99 63 70)(51 91 55 71)(52 92 56 72)(53 93 57 64)(54 94 58 65)
(1 131 46 114 42 102 59 125)(2 132 47 115 43 103 60 126)(3 133 48 116 44 104 61 118)(4 134 49 117 45 105 62 119)(5 135 50 109 37 106 63 120)(6 127 51 110 38 107 55 121)(7 128 52 111 39 108 56 122)(8 129 53 112 40 100 57 123)(9 130 54 113 41 101 58 124)(10 86 34 72 20 75 144 92)(11 87 35 64 21 76 136 93)(12 88 36 65 22 77 137 94)(13 89 28 66 23 78 138 95)(14 90 29 67 24 79 139 96)(15 82 30 68 25 80 140 97)(16 83 31 69 26 81 141 98)(17 84 32 70 27 73 142 99)(18 85 33 71 19 74 143 91)
(1 125)(2 126)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 86)(11 87)(12 88)(13 89)(14 90)(15 82)(16 83)(17 84)(18 85)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 73)(28 95)(29 96)(30 97)(31 98)(32 99)(33 91)(34 92)(35 93)(36 94)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 100)(54 101)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
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), (1,78,42,89)(2,79,43,90)(3,80,44,82)(4,81,45,83)(5,73,37,84)(6,74,38,85)(7,75,39,86)(8,76,40,87)(9,77,41,88)(10,122,20,111)(11,123,21,112)(12,124,22,113)(13,125,23,114)(14,126,24,115)(15,118,25,116)(16,119,26,117)(17,120,27,109)(18,121,19,110)(28,131,138,102)(29,132,139,103)(30,133,140,104)(31,134,141,105)(32,135,142,106)(33,127,143,107)(34,128,144,108)(35,129,136,100)(36,130,137,101)(46,95,59,66)(47,96,60,67)(48,97,61,68)(49,98,62,69)(50,99,63,70)(51,91,55,71)(52,92,56,72)(53,93,57,64)(54,94,58,65), (1,131,46,114,42,102,59,125)(2,132,47,115,43,103,60,126)(3,133,48,116,44,104,61,118)(4,134,49,117,45,105,62,119)(5,135,50,109,37,106,63,120)(6,127,51,110,38,107,55,121)(7,128,52,111,39,108,56,122)(8,129,53,112,40,100,57,123)(9,130,54,113,41,101,58,124)(10,86,34,72,20,75,144,92)(11,87,35,64,21,76,136,93)(12,88,36,65,22,77,137,94)(13,89,28,66,23,78,138,95)(14,90,29,67,24,79,139,96)(15,82,30,68,25,80,140,97)(16,83,31,69,26,81,141,98)(17,84,32,70,27,73,142,99)(18,85,33,71,19,74,143,91), (1,125)(2,126)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,86)(11,87)(12,88)(13,89)(14,90)(15,82)(16,83)(17,84)(18,85)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,73)(28,95)(29,96)(30,97)(31,98)(32,99)(33,91)(34,92)(35,93)(36,94)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,100)(54,101)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)>;
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), (1,78,42,89)(2,79,43,90)(3,80,44,82)(4,81,45,83)(5,73,37,84)(6,74,38,85)(7,75,39,86)(8,76,40,87)(9,77,41,88)(10,122,20,111)(11,123,21,112)(12,124,22,113)(13,125,23,114)(14,126,24,115)(15,118,25,116)(16,119,26,117)(17,120,27,109)(18,121,19,110)(28,131,138,102)(29,132,139,103)(30,133,140,104)(31,134,141,105)(32,135,142,106)(33,127,143,107)(34,128,144,108)(35,129,136,100)(36,130,137,101)(46,95,59,66)(47,96,60,67)(48,97,61,68)(49,98,62,69)(50,99,63,70)(51,91,55,71)(52,92,56,72)(53,93,57,64)(54,94,58,65), (1,131,46,114,42,102,59,125)(2,132,47,115,43,103,60,126)(3,133,48,116,44,104,61,118)(4,134,49,117,45,105,62,119)(5,135,50,109,37,106,63,120)(6,127,51,110,38,107,55,121)(7,128,52,111,39,108,56,122)(8,129,53,112,40,100,57,123)(9,130,54,113,41,101,58,124)(10,86,34,72,20,75,144,92)(11,87,35,64,21,76,136,93)(12,88,36,65,22,77,137,94)(13,89,28,66,23,78,138,95)(14,90,29,67,24,79,139,96)(15,82,30,68,25,80,140,97)(16,83,31,69,26,81,141,98)(17,84,32,70,27,73,142,99)(18,85,33,71,19,74,143,91), (1,125)(2,126)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,86)(11,87)(12,88)(13,89)(14,90)(15,82)(16,83)(17,84)(18,85)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,73)(28,95)(29,96)(30,97)(31,98)(32,99)(33,91)(34,92)(35,93)(36,94)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,100)(54,101)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144) );
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)], [(1,78,42,89),(2,79,43,90),(3,80,44,82),(4,81,45,83),(5,73,37,84),(6,74,38,85),(7,75,39,86),(8,76,40,87),(9,77,41,88),(10,122,20,111),(11,123,21,112),(12,124,22,113),(13,125,23,114),(14,126,24,115),(15,118,25,116),(16,119,26,117),(17,120,27,109),(18,121,19,110),(28,131,138,102),(29,132,139,103),(30,133,140,104),(31,134,141,105),(32,135,142,106),(33,127,143,107),(34,128,144,108),(35,129,136,100),(36,130,137,101),(46,95,59,66),(47,96,60,67),(48,97,61,68),(49,98,62,69),(50,99,63,70),(51,91,55,71),(52,92,56,72),(53,93,57,64),(54,94,58,65)], [(1,131,46,114,42,102,59,125),(2,132,47,115,43,103,60,126),(3,133,48,116,44,104,61,118),(4,134,49,117,45,105,62,119),(5,135,50,109,37,106,63,120),(6,127,51,110,38,107,55,121),(7,128,52,111,39,108,56,122),(8,129,53,112,40,100,57,123),(9,130,54,113,41,101,58,124),(10,86,34,72,20,75,144,92),(11,87,35,64,21,76,136,93),(12,88,36,65,22,77,137,94),(13,89,28,66,23,78,138,95),(14,90,29,67,24,79,139,96),(15,82,30,68,25,80,140,97),(16,83,31,69,26,81,141,98),(17,84,32,70,27,73,142,99),(18,85,33,71,19,74,143,91)], [(1,125),(2,126),(3,118),(4,119),(5,120),(6,121),(7,122),(8,123),(9,124),(10,86),(11,87),(12,88),(13,89),(14,90),(15,82),(16,83),(17,84),(18,85),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,73),(28,95),(29,96),(30,97),(31,98),(32,99),(33,91),(34,92),(35,93),(36,94),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,100),(54,101),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18X | 24A | ··· | 24H | 36A | ··· | 36L | 36M | ··· | 36R | 36S | ··· | 36AD | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | C18 | D4 | D4 | C3×D4 | C3×D4 | C4○D8 | D4×C9 | D4×C9 | C3×C4○D8 | C9×C4○D8 |
| kernel | C9×C4○D8 | C2×C72 | C9×D8 | C9×SD16 | C9×Q16 | C9×C4○D4 | C3×C4○D8 | C2×C24 | C3×D8 | C3×SD16 | C3×Q16 | C3×C4○D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C36 | C2×C18 | C12 | C2×C6 | C9 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 6 | 6 | 6 | 12 | 6 | 12 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 8 | 24 |
Matrix representation of C9×C4○D8 ►in GL2(𝔽73) generated by
| 37 | 0 |
| 0 | 37 |
| 27 | 0 |
| 0 | 27 |
| 57 | 16 |
| 57 | 57 |
| 57 | 16 |
| 16 | 16 |
G:=sub<GL(2,GF(73))| [37,0,0,37],[27,0,0,27],[57,57,16,57],[57,16,16,16] >;
C9×C4○D8 in GAP, Magma, Sage, TeX
C_9\times C_4\circ D_8
% in TeX
G:=Group("C9xC4oD8"); // GroupNames label
G:=SmallGroup(288,185);
// by ID
G=gap.SmallGroup(288,185);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,772,192,5884,2951,242]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations