direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22:C4xC18, C23:3C36, C24.2C18, C6.64(C6xD4), C2.1(D4xC18), (C22xC36):3C2, (C22xC18):3C4, C22:3(C2xC36), (C22xC4):3C18, C18.64(C2xD4), (C2xC18).50D4, (C23xC6).8C6, (C2xC36):11C22, C2.1(C22xC36), (C23xC18).1C2, C22.12(D4xC9), (C22xC6).10C12, (C22xC12).11C6, C23.10(C2xC18), (C2xC18).70C23, C6.29(C22xC12), C18.29(C22xC4), C22.3(C22xC18), (C22xC18).24C22, C3.(C6xC22:C4), (C2xC4):3(C2xC18), (C2xC18):7(C2xC4), (C6xC22:C4).C3, (C2xC6).59(C3xD4), (C2xC6).26(C2xC12), (C2xC12).79(C2xC6), C6.27(C3xC22:C4), (C3xC22:C4).13C6, (C22xC6).42(C2xC6), (C2xC6).75(C22xC6), SmallGroup(288,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22:C4xC18
G = < a,b,c,d | a18=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 282 in 198 conjugacy classes, 114 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2xC4, C2xC4, C23, C23, C23, C9, C12, C2xC6, C2xC6, C2xC6, C22:C4, C22xC4, C24, C18, C18, C18, C2xC12, C2xC12, C22xC6, C22xC6, C22xC6, C2xC22:C4, C36, C2xC18, C2xC18, C2xC18, C3xC22:C4, C22xC12, C23xC6, C2xC36, C2xC36, C22xC18, C22xC18, C22xC18, C6xC22:C4, C9xC22:C4, C22xC36, C23xC18, C22:C4xC18
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, C23, C9, C12, C2xC6, C22:C4, C22xC4, C2xD4, C18, C2xC12, C3xD4, C22xC6, C2xC22:C4, C36, C2xC18, C3xC22:C4, C22xC12, C6xD4, C2xC36, D4xC9, C22xC18, C6xC22:C4, C9xC22:C4, C22xC36, D4xC18, C22:C4xC18
►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)
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 109)(10 110)(11 111)(12 112)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 90)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 91)(71 92)(72 93)
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 141)(10 142)(11 143)(12 144)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 55)(34 56)(35 57)(36 58)(37 125)(38 126)(39 109)(40 110)(41 111)(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)
(1 100 119 21)(2 101 120 22)(3 102 121 23)(4 103 122 24)(5 104 123 25)(6 105 124 26)(7 106 125 27)(8 107 126 28)(9 108 109 29)(10 91 110 30)(11 92 111 31)(12 93 112 32)(13 94 113 33)(14 95 114 34)(15 96 115 35)(16 97 116 36)(17 98 117 19)(18 99 118 20)(37 67 139 80)(38 68 140 81)(39 69 141 82)(40 70 142 83)(41 71 143 84)(42 72 144 85)(43 55 127 86)(44 56 128 87)(45 57 129 88)(46 58 130 89)(47 59 131 90)(48 60 132 73)(49 61 133 74)(50 62 134 75)(51 63 135 76)(52 64 136 77)(53 65 137 78)(54 66 138 79)
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), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,109)(10,110)(11,111)(12,112)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,90)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,91)(71,92)(72,93), (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,141)(10,142)(11,143)(12,144)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,55)(34,56)(35,57)(36,58)(37,125)(38,126)(39,109)(40,110)(41,111)(42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98), (1,100,119,21)(2,101,120,22)(3,102,121,23)(4,103,122,24)(5,104,123,25)(6,105,124,26)(7,106,125,27)(8,107,126,28)(9,108,109,29)(10,91,110,30)(11,92,111,31)(12,93,112,32)(13,94,113,33)(14,95,114,34)(15,96,115,35)(16,97,116,36)(17,98,117,19)(18,99,118,20)(37,67,139,80)(38,68,140,81)(39,69,141,82)(40,70,142,83)(41,71,143,84)(42,72,144,85)(43,55,127,86)(44,56,128,87)(45,57,129,88)(46,58,130,89)(47,59,131,90)(48,60,132,73)(49,61,133,74)(50,62,134,75)(51,63,135,76)(52,64,136,77)(53,65,137,78)(54,66,138,79)>;
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), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,109)(10,110)(11,111)(12,112)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,90)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,91)(71,92)(72,93), (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,141)(10,142)(11,143)(12,144)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,55)(34,56)(35,57)(36,58)(37,125)(38,126)(39,109)(40,110)(41,111)(42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98), (1,100,119,21)(2,101,120,22)(3,102,121,23)(4,103,122,24)(5,104,123,25)(6,105,124,26)(7,106,125,27)(8,107,126,28)(9,108,109,29)(10,91,110,30)(11,92,111,31)(12,93,112,32)(13,94,113,33)(14,95,114,34)(15,96,115,35)(16,97,116,36)(17,98,117,19)(18,99,118,20)(37,67,139,80)(38,68,140,81)(39,69,141,82)(40,70,142,83)(41,71,143,84)(42,72,144,85)(43,55,127,86)(44,56,128,87)(45,57,129,88)(46,58,130,89)(47,59,131,90)(48,60,132,73)(49,61,133,74)(50,62,134,75)(51,63,135,76)(52,64,136,77)(53,65,137,78)(54,66,138,79) );
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)], [(1,119),(2,120),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,109),(10,110),(11,111),(12,112),(13,113),(14,114),(15,115),(16,116),(17,117),(18,118),(19,90),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,139),(38,140),(39,141),(40,142),(41,143),(42,144),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,91),(71,92),(72,93)], [(1,133),(2,134),(3,135),(4,136),(5,137),(6,138),(7,139),(8,140),(9,141),(10,142),(11,143),(12,144),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,55),(34,56),(35,57),(36,58),(37,125),(38,126),(39,109),(40,110),(41,111),(42,112),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,121),(52,122),(53,123),(54,124),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98)], [(1,100,119,21),(2,101,120,22),(3,102,121,23),(4,103,122,24),(5,104,123,25),(6,105,124,26),(7,106,125,27),(8,107,126,28),(9,108,109,29),(10,91,110,30),(11,92,111,31),(12,93,112,32),(13,94,113,33),(14,95,114,34),(15,96,115,35),(16,97,116,36),(17,98,117,19),(18,99,118,20),(37,67,139,80),(38,68,140,81),(39,69,141,82),(40,70,142,83),(41,71,143,84),(42,72,144,85),(43,55,127,86),(44,56,128,87),(45,57,129,88),(46,58,130,89),(47,59,131,90),(48,60,132,73),(49,61,133,74),(50,62,134,75),(51,63,135,76),(52,64,136,77),(53,65,137,78),(54,66,138,79)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6V | 9A | ··· | 9F | 12A | ··· | 12P | 18A | ··· | 18AP | 18AQ | ··· | 18BN | 36A | ··· | 36AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C9 | C12 | C18 | C18 | C18 | C36 | D4 | C3xD4 | D4xC9 |
| kernel | C22:C4xC18 | C9xC22:C4 | C22xC36 | C23xC18 | C6xC22:C4 | C22xC18 | C3xC22:C4 | C22xC12 | C23xC6 | C2xC22:C4 | C22xC6 | C22:C4 | C22xC4 | C24 | C23 | C2xC18 | C2xC6 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 2 | 6 | 16 | 24 | 12 | 6 | 48 | 4 | 8 | 24 |
Matrix representation of C22:C4xC18 ►in GL4(F37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 30 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 6 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 25 |
| 0 | 0 | 35 | 30 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,30,1],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[6,0,0,0,0,1,0,0,0,0,7,35,0,0,25,30] >;
C22:C4xC18 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{18} % in TeX
G:=Group("C2^2:C4xC18"); // GroupNames label
G:=SmallGroup(288,165);
// by ID
G=gap.SmallGroup(288,165);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,365,360]);
// Polycyclic
G:=Group<a,b,c,d|a^18=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations