direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C18, C72⋊14C22, C23.4C36, C36.54C23, C8⋊4(C2×C18), (C2×C8)⋊6C18, (C2×C72)⋊14C2, (C2×C4).6C36, C4.9(C2×C36), C3.(C6×M4(2)), (C2×C36).15C4, (C2×C24).30C6, C36.47(C2×C4), C24.38(C2×C6), (C6×M4(2)).C3, C12.54(C2×C12), (C2×C12).22C12, C22.6(C2×C36), (C22×C4).8C18, (C22×C18).4C4, C2.6(C22×C36), (C22×C36).16C2, C18.34(C22×C4), (C22×C12).30C6, C12.65(C22×C6), (C22×C6).11C12, C4.11(C22×C18), C6.34(C22×C12), C6.10(C3×M4(2)), (C2×C36).136C22, (C3×M4(2)).11C6, (C2×C18).23(C2×C4), (C2×C4).24(C2×C18), (C2×C6).28(C2×C12), (C2×C12).142(C2×C6), SmallGroup(288,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C18
G = < a,b,c | a18=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Subgroups: 114 in 102 conjugacy classes, 90 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C18, C18, C18, C24, C2×C12, C2×C12, C22×C6, C2×M4(2), C36, C36, C2×C18, C2×C18, C2×C18, C2×C24, C3×M4(2), C22×C12, C72, C2×C36, C2×C36, C22×C18, C6×M4(2), C2×C72, C9×M4(2), C22×C36, M4(2)×C18
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C9, C12, C2×C6, M4(2), C22×C4, C18, C2×C12, C22×C6, C2×M4(2), C36, C2×C18, C3×M4(2), C22×C12, C2×C36, C22×C18, C6×M4(2), C9×M4(2), C22×C36, M4(2)×C18
►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 21 120 136 61 73 51 103)(2 22 121 137 62 74 52 104)(3 23 122 138 63 75 53 105)(4 24 123 139 64 76 54 106)(5 25 124 140 65 77 37 107)(6 26 125 141 66 78 38 108)(7 27 126 142 67 79 39 91)(8 28 109 143 68 80 40 92)(9 29 110 144 69 81 41 93)(10 30 111 127 70 82 42 94)(11 31 112 128 71 83 43 95)(12 32 113 129 72 84 44 96)(13 33 114 130 55 85 45 97)(14 34 115 131 56 86 46 98)(15 35 116 132 57 87 47 99)(16 36 117 133 58 88 48 100)(17 19 118 134 59 89 49 101)(18 20 119 135 60 90 50 102)
(19 89)(20 90)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(91 142)(92 143)(93 144)(94 127)(95 128)(96 129)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 141)
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), (1,21,120,136,61,73,51,103)(2,22,121,137,62,74,52,104)(3,23,122,138,63,75,53,105)(4,24,123,139,64,76,54,106)(5,25,124,140,65,77,37,107)(6,26,125,141,66,78,38,108)(7,27,126,142,67,79,39,91)(8,28,109,143,68,80,40,92)(9,29,110,144,69,81,41,93)(10,30,111,127,70,82,42,94)(11,31,112,128,71,83,43,95)(12,32,113,129,72,84,44,96)(13,33,114,130,55,85,45,97)(14,34,115,131,56,86,46,98)(15,35,116,132,57,87,47,99)(16,36,117,133,58,88,48,100)(17,19,118,134,59,89,49,101)(18,20,119,135,60,90,50,102), (19,89)(20,90)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(91,142)(92,143)(93,144)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,141)>;
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), (1,21,120,136,61,73,51,103)(2,22,121,137,62,74,52,104)(3,23,122,138,63,75,53,105)(4,24,123,139,64,76,54,106)(5,25,124,140,65,77,37,107)(6,26,125,141,66,78,38,108)(7,27,126,142,67,79,39,91)(8,28,109,143,68,80,40,92)(9,29,110,144,69,81,41,93)(10,30,111,127,70,82,42,94)(11,31,112,128,71,83,43,95)(12,32,113,129,72,84,44,96)(13,33,114,130,55,85,45,97)(14,34,115,131,56,86,46,98)(15,35,116,132,57,87,47,99)(16,36,117,133,58,88,48,100)(17,19,118,134,59,89,49,101)(18,20,119,135,60,90,50,102), (19,89)(20,90)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(91,142)(92,143)(93,144)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,141) );
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)], [(1,21,120,136,61,73,51,103),(2,22,121,137,62,74,52,104),(3,23,122,138,63,75,53,105),(4,24,123,139,64,76,54,106),(5,25,124,140,65,77,37,107),(6,26,125,141,66,78,38,108),(7,27,126,142,67,79,39,91),(8,28,109,143,68,80,40,92),(9,29,110,144,69,81,41,93),(10,30,111,127,70,82,42,94),(11,31,112,128,71,83,43,95),(12,32,113,129,72,84,44,96),(13,33,114,130,55,85,45,97),(14,34,115,131,56,86,46,98),(15,35,116,132,57,87,47,99),(16,36,117,133,58,88,48,100),(17,19,118,134,59,89,49,101),(18,20,119,135,60,90,50,102)], [(19,89),(20,90),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(91,142),(92,143),(93,144),(94,127),(95,128),(96,129),(97,130),(98,131),(99,132),(100,133),(101,134),(102,135),(103,136),(104,137),(105,138),(106,139),(107,140),(108,141)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 9A | ··· | 9F | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 18A | ··· | 18R | 18S | ··· | 18AD | 24A | ··· | 24P | 36A | ··· | 36X | 36Y | ··· | 36AJ | 72A | ··· | 72AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 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 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C18 | C36 | C36 | M4(2) | C3×M4(2) | C9×M4(2) |
| kernel | M4(2)×C18 | C2×C72 | C9×M4(2) | C22×C36 | C6×M4(2) | C2×C36 | C22×C18 | C2×C24 | C3×M4(2) | C22×C12 | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C18 | C6 | C2 |
| # reps | 1 | 2 | 4 | 1 | 2 | 6 | 2 | 4 | 8 | 2 | 6 | 12 | 4 | 12 | 24 | 6 | 36 | 12 | 4 | 8 | 24 |
Matrix representation of M4(2)×C18 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 72 | 0 | 0 |
| 0 | 0 | 2 |
| 0 | 23 | 0 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 72 |
G:=sub<GL(3,GF(73))| [72,0,0,0,4,0,0,0,4],[72,0,0,0,0,23,0,2,0],[72,0,0,0,1,0,0,0,72] >;
M4(2)×C18 in GAP, Magma, Sage, TeX
M_4(2)\times C_{18} % in TeX
G:=Group("M4(2)xC18"); // GroupNames label
G:=SmallGroup(288,180);
// by ID
G=gap.SmallGroup(288,180);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,168,2045,192,242]);
// Polycyclic
G:=Group<a,b,c|a^18=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations