direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×C8.C22, Q16⋊2C18, C36.64D4, SD16⋊2C18, M4(2)⋊2C18, C36.49C23, C72.13C22, C8.(C2×C18), (C9×Q16)⋊6C2, (C2×Q8)⋊6C18, C4.15(D4×C9), C6.79(C6×D4), C24.13(C2×C6), (Q8×C18)⋊11C2, C4○D4.4C18, (C9×SD16)⋊6C2, D4.3(C2×C18), C2.16(D4×C18), C12.74(C3×D4), C18.79(C2×D4), (C2×C18).26D4, Q8.6(C2×C18), (C6×Q8).21C6, (C3×Q16).5C6, C22.6(D4×C9), (C9×M4(2))⋊6C2, C4.6(C22×C18), (C3×SD16).2C6, (C2×C36).68C22, C12.49(C22×C6), (D4×C9).13C22, (C3×M4(2)).2C6, (Q8×C9).14C22, C3.(C3×C8.C22), (C2×C4).5(C2×C18), (C9×C4○D4).5C2, (C2×C6).30(C3×D4), (C3×C8.C22).C3, (C2×C12).66(C2×C6), (C3×C4○D4).15C6, (C3×D4).15(C2×C6), (C3×Q8).28(C2×C6), SmallGroup(288,187)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C8.C22
G = < a,b,c,d | a9=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
Subgroups: 126 in 90 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C9, C12, C12, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C18, C18, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C8.C22, C36, C36, C2×C18, C2×C18, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C72, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, Q8×C9, Q8×C9, C3×C8.C22, C9×M4(2), C9×SD16, C9×Q16, Q8×C18, C9×C4○D4, C9×C8.C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C18, C3×D4, C22×C6, C8.C22, C2×C18, C6×D4, D4×C9, C22×C18, C3×C8.C22, D4×C18, C9×C8.C22
►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)
(1 114 59 131 42 125 46 102)(2 115 60 132 43 126 47 103)(3 116 61 133 44 118 48 104)(4 117 62 134 45 119 49 105)(5 109 63 135 37 120 50 106)(6 110 55 127 38 121 51 107)(7 111 56 128 39 122 52 108)(8 112 57 129 40 123 53 100)(9 113 58 130 41 124 54 101)(10 92 144 75 20 72 34 86)(11 93 136 76 21 64 35 87)(12 94 137 77 22 65 36 88)(13 95 138 78 23 66 28 89)(14 96 139 79 24 67 29 90)(15 97 140 80 25 68 30 82)(16 98 141 81 26 69 31 83)(17 99 142 73 27 70 32 84)(18 91 143 74 19 71 33 85)
(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 143)(20 144)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(46 59)(47 60)(48 61)(49 62)(50 63)(51 55)(52 56)(53 57)(54 58)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 82)(81 83)(100 123)(101 124)(102 125)(103 126)(104 118)(105 119)(106 120)(107 121)(108 122)(109 135)(110 127)(111 128)(112 129)(113 130)(114 131)(115 132)(116 133)(117 134)
(1 89)(2 90)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 122)(11 123)(12 124)(13 125)(14 126)(15 118)(16 119)(17 120)(18 121)(19 110)(20 111)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 109)(28 131)(29 132)(30 133)(31 134)(32 135)(33 127)(34 128)(35 129)(36 130)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 64)(54 65)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
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), (1,114,59,131,42,125,46,102)(2,115,60,132,43,126,47,103)(3,116,61,133,44,118,48,104)(4,117,62,134,45,119,49,105)(5,109,63,135,37,120,50,106)(6,110,55,127,38,121,51,107)(7,111,56,128,39,122,52,108)(8,112,57,129,40,123,53,100)(9,113,58,130,41,124,54,101)(10,92,144,75,20,72,34,86)(11,93,136,76,21,64,35,87)(12,94,137,77,22,65,36,88)(13,95,138,78,23,66,28,89)(14,96,139,79,24,67,29,90)(15,97,140,80,25,68,30,82)(16,98,141,81,26,69,31,83)(17,99,142,73,27,70,32,84)(18,91,143,74,19,71,33,85), (10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,143)(20,144)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(46,59)(47,60)(48,61)(49,62)(50,63)(51,55)(52,56)(53,57)(54,58)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,82)(81,83)(100,123)(101,124)(102,125)(103,126)(104,118)(105,119)(106,120)(107,121)(108,122)(109,135)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134), (1,89)(2,90)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,122)(11,123)(12,124)(13,125)(14,126)(15,118)(16,119)(17,120)(18,121)(19,110)(20,111)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,109)(28,131)(29,132)(30,133)(31,134)(32,135)(33,127)(34,128)(35,129)(36,130)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,64)(54,65)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)>;
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), (1,114,59,131,42,125,46,102)(2,115,60,132,43,126,47,103)(3,116,61,133,44,118,48,104)(4,117,62,134,45,119,49,105)(5,109,63,135,37,120,50,106)(6,110,55,127,38,121,51,107)(7,111,56,128,39,122,52,108)(8,112,57,129,40,123,53,100)(9,113,58,130,41,124,54,101)(10,92,144,75,20,72,34,86)(11,93,136,76,21,64,35,87)(12,94,137,77,22,65,36,88)(13,95,138,78,23,66,28,89)(14,96,139,79,24,67,29,90)(15,97,140,80,25,68,30,82)(16,98,141,81,26,69,31,83)(17,99,142,73,27,70,32,84)(18,91,143,74,19,71,33,85), (10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,143)(20,144)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(46,59)(47,60)(48,61)(49,62)(50,63)(51,55)(52,56)(53,57)(54,58)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,82)(81,83)(100,123)(101,124)(102,125)(103,126)(104,118)(105,119)(106,120)(107,121)(108,122)(109,135)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134), (1,89)(2,90)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,122)(11,123)(12,124)(13,125)(14,126)(15,118)(16,119)(17,120)(18,121)(19,110)(20,111)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,109)(28,131)(29,132)(30,133)(31,134)(32,135)(33,127)(34,128)(35,129)(36,130)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,64)(54,65)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144) );
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)], [(1,114,59,131,42,125,46,102),(2,115,60,132,43,126,47,103),(3,116,61,133,44,118,48,104),(4,117,62,134,45,119,49,105),(5,109,63,135,37,120,50,106),(6,110,55,127,38,121,51,107),(7,111,56,128,39,122,52,108),(8,112,57,129,40,123,53,100),(9,113,58,130,41,124,54,101),(10,92,144,75,20,72,34,86),(11,93,136,76,21,64,35,87),(12,94,137,77,22,65,36,88),(13,95,138,78,23,66,28,89),(14,96,139,79,24,67,29,90),(15,97,140,80,25,68,30,82),(16,98,141,81,26,69,31,83),(17,99,142,73,27,70,32,84),(18,91,143,74,19,71,33,85)], [(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,143),(20,144),(21,136),(22,137),(23,138),(24,139),(25,140),(26,141),(27,142),(46,59),(47,60),(48,61),(49,62),(50,63),(51,55),(52,56),(53,57),(54,58),(73,84),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,82),(81,83),(100,123),(101,124),(102,125),(103,126),(104,118),(105,119),(106,120),(107,121),(108,122),(109,135),(110,127),(111,128),(112,129),(113,130),(114,131),(115,132),(116,133),(117,134)], [(1,89),(2,90),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,122),(11,123),(12,124),(13,125),(14,126),(15,118),(16,119),(17,120),(18,121),(19,110),(20,111),(21,112),(22,113),(23,114),(24,115),(25,116),(26,117),(27,109),(28,131),(29,132),(30,133),(31,134),(32,135),(33,127),(34,128),(35,129),(36,130),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,64),(54,65),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18R | 24A | 24B | 24C | 24D | 36A | ··· | 36L | 36M | ··· | 36AD | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
99 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 | 4 | 4 | 4 |
| 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 | D4×C9 | D4×C9 | C8.C22 | C3×C8.C22 | C9×C8.C22 |
| kernel | C9×C8.C22 | C9×M4(2) | C9×SD16 | C9×Q16 | Q8×C18 | C9×C4○D4 | C3×C8.C22 | C3×M4(2) | C3×SD16 | C3×Q16 | C6×Q8 | C3×C4○D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C36 | C2×C18 | C12 | C2×C6 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 6 | 6 | 12 | 12 | 6 | 6 | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of C9×C8.C22 ►in GL4(𝔽73) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 18 | 11 | 62 | 18 |
| 14 | 43 | 17 | 62 |
| 19 | 19 | 30 | 62 |
| 54 | 19 | 59 | 55 |
| 1 | 0 | 10 | 0 |
| 0 | 72 | 0 | 63 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 64 | 17 | 56 |
| 64 | 10 | 56 | 17 |
| 71 | 0 | 63 | 9 |
| 0 | 71 | 9 | 63 |
G:=sub<GL(4,GF(73))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[18,14,19,54,11,43,19,19,62,17,30,59,18,62,62,55],[1,0,0,0,0,72,0,0,10,0,72,0,0,63,0,1],[10,64,71,0,64,10,0,71,17,56,63,9,56,17,9,63] >;
C9×C8.C22 in GAP, Magma, Sage, TeX
C_9\times C_8.C_2^2
% in TeX
G:=Group("C9xC8.C2^2"); // GroupNames label
G:=SmallGroup(288,187);
// by ID
G=gap.SmallGroup(288,187);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,1016,3110,192,5884,2951,242]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations