direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C32×M5(2), C48⋊7C6, C12.6C24, C62.5C8, C8.8C62, C24.11C12, C4.(C3×C24), C16⋊3(C3×C6), (C3×C48)⋊11C2, C8.2(C3×C12), (C2×C6).4C24, C2.3(C6×C24), C22.(C3×C24), (C2×C24).33C6, (C6×C24).28C2, (C6×C12).34C4, C24.42(C2×C6), (C3×C24).15C4, (C3×C12).13C8, C6.16(C2×C24), C4.10(C6×C12), C12.62(C2×C12), (C2×C12).24C12, (C3×C24).76C22, (C2×C8).8(C3×C6), (C2×C4).5(C3×C12), (C3×C6).49(C2×C8), (C3×C12).146(C2×C4), SmallGroup(288,328)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×M5(2)
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >
Subgroups: 84 in 78 conjugacy classes, 72 normal (20 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C2×C6, C16, C2×C8, C3×C6, C3×C6, C24, C2×C12, M5(2), C3×C12, C62, C48, C2×C24, C3×C24, C6×C12, C3×M5(2), C3×C48, C6×C24, C32×M5(2)
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C24, C2×C12, M5(2), C3×C12, C62, C2×C24, C3×C24, C6×C12, C3×M5(2), C6×C24, C32×M5(2)
►On 144 points
Generators in S144
(1 34 90)(2 35 91)(3 36 92)(4 37 93)(5 38 94)(6 39 95)(7 40 96)(8 41 81)(9 42 82)(10 43 83)(11 44 84)(12 45 85)(13 46 86)(14 47 87)(15 48 88)(16 33 89)(17 101 68)(18 102 69)(19 103 70)(20 104 71)(21 105 72)(22 106 73)(23 107 74)(24 108 75)(25 109 76)(26 110 77)(27 111 78)(28 112 79)(29 97 80)(30 98 65)(31 99 66)(32 100 67)(49 113 143)(50 114 144)(51 115 129)(52 116 130)(53 117 131)(54 118 132)(55 119 133)(56 120 134)(57 121 135)(58 122 136)(59 123 137)(60 124 138)(61 125 139)(62 126 140)(63 127 141)(64 128 142)
(1 68 118)(2 69 119)(3 70 120)(4 71 121)(5 72 122)(6 73 123)(7 74 124)(8 75 125)(9 76 126)(10 77 127)(11 78 128)(12 79 113)(13 80 114)(14 65 115)(15 66 116)(16 67 117)(17 132 34)(18 133 35)(19 134 36)(20 135 37)(21 136 38)(22 137 39)(23 138 40)(24 139 41)(25 140 42)(26 141 43)(27 142 44)(28 143 45)(29 144 46)(30 129 47)(31 130 48)(32 131 33)(49 85 112)(50 86 97)(51 87 98)(52 88 99)(53 89 100)(54 90 101)(55 91 102)(56 92 103)(57 93 104)(58 94 105)(59 95 106)(60 96 107)(61 81 108)(62 82 109)(63 83 110)(64 84 111)
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(98 106)(100 108)(102 110)(104 112)(113 121)(115 123)(117 125)(119 127)(129 137)(131 139)(133 141)(135 143)
G:=sub<Sym(144)| (1,34,90)(2,35,91)(3,36,92)(4,37,93)(5,38,94)(6,39,95)(7,40,96)(8,41,81)(9,42,82)(10,43,83)(11,44,84)(12,45,85)(13,46,86)(14,47,87)(15,48,88)(16,33,89)(17,101,68)(18,102,69)(19,103,70)(20,104,71)(21,105,72)(22,106,73)(23,107,74)(24,108,75)(25,109,76)(26,110,77)(27,111,78)(28,112,79)(29,97,80)(30,98,65)(31,99,66)(32,100,67)(49,113,143)(50,114,144)(51,115,129)(52,116,130)(53,117,131)(54,118,132)(55,119,133)(56,120,134)(57,121,135)(58,122,136)(59,123,137)(60,124,138)(61,125,139)(62,126,140)(63,127,141)(64,128,142), (1,68,118)(2,69,119)(3,70,120)(4,71,121)(5,72,122)(6,73,123)(7,74,124)(8,75,125)(9,76,126)(10,77,127)(11,78,128)(12,79,113)(13,80,114)(14,65,115)(15,66,116)(16,67,117)(17,132,34)(18,133,35)(19,134,36)(20,135,37)(21,136,38)(22,137,39)(23,138,40)(24,139,41)(25,140,42)(26,141,43)(27,142,44)(28,143,45)(29,144,46)(30,129,47)(31,130,48)(32,131,33)(49,85,112)(50,86,97)(51,87,98)(52,88,99)(53,89,100)(54,90,101)(55,91,102)(56,92,103)(57,93,104)(58,94,105)(59,95,106)(60,96,107)(61,81,108)(62,82,109)(63,83,110)(64,84,111), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(98,106)(100,108)(102,110)(104,112)(113,121)(115,123)(117,125)(119,127)(129,137)(131,139)(133,141)(135,143)>;
G:=Group( (1,34,90)(2,35,91)(3,36,92)(4,37,93)(5,38,94)(6,39,95)(7,40,96)(8,41,81)(9,42,82)(10,43,83)(11,44,84)(12,45,85)(13,46,86)(14,47,87)(15,48,88)(16,33,89)(17,101,68)(18,102,69)(19,103,70)(20,104,71)(21,105,72)(22,106,73)(23,107,74)(24,108,75)(25,109,76)(26,110,77)(27,111,78)(28,112,79)(29,97,80)(30,98,65)(31,99,66)(32,100,67)(49,113,143)(50,114,144)(51,115,129)(52,116,130)(53,117,131)(54,118,132)(55,119,133)(56,120,134)(57,121,135)(58,122,136)(59,123,137)(60,124,138)(61,125,139)(62,126,140)(63,127,141)(64,128,142), (1,68,118)(2,69,119)(3,70,120)(4,71,121)(5,72,122)(6,73,123)(7,74,124)(8,75,125)(9,76,126)(10,77,127)(11,78,128)(12,79,113)(13,80,114)(14,65,115)(15,66,116)(16,67,117)(17,132,34)(18,133,35)(19,134,36)(20,135,37)(21,136,38)(22,137,39)(23,138,40)(24,139,41)(25,140,42)(26,141,43)(27,142,44)(28,143,45)(29,144,46)(30,129,47)(31,130,48)(32,131,33)(49,85,112)(50,86,97)(51,87,98)(52,88,99)(53,89,100)(54,90,101)(55,91,102)(56,92,103)(57,93,104)(58,94,105)(59,95,106)(60,96,107)(61,81,108)(62,82,109)(63,83,110)(64,84,111), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(98,106)(100,108)(102,110)(104,112)(113,121)(115,123)(117,125)(119,127)(129,137)(131,139)(133,141)(135,143) );
G=PermutationGroup([[(1,34,90),(2,35,91),(3,36,92),(4,37,93),(5,38,94),(6,39,95),(7,40,96),(8,41,81),(9,42,82),(10,43,83),(11,44,84),(12,45,85),(13,46,86),(14,47,87),(15,48,88),(16,33,89),(17,101,68),(18,102,69),(19,103,70),(20,104,71),(21,105,72),(22,106,73),(23,107,74),(24,108,75),(25,109,76),(26,110,77),(27,111,78),(28,112,79),(29,97,80),(30,98,65),(31,99,66),(32,100,67),(49,113,143),(50,114,144),(51,115,129),(52,116,130),(53,117,131),(54,118,132),(55,119,133),(56,120,134),(57,121,135),(58,122,136),(59,123,137),(60,124,138),(61,125,139),(62,126,140),(63,127,141),(64,128,142)], [(1,68,118),(2,69,119),(3,70,120),(4,71,121),(5,72,122),(6,73,123),(7,74,124),(8,75,125),(9,76,126),(10,77,127),(11,78,128),(12,79,113),(13,80,114),(14,65,115),(15,66,116),(16,67,117),(17,132,34),(18,133,35),(19,134,36),(20,135,37),(21,136,38),(22,137,39),(23,138,40),(24,139,41),(25,140,42),(26,141,43),(27,142,44),(28,143,45),(29,144,46),(30,129,47),(31,130,48),(32,131,33),(49,85,112),(50,86,97),(51,87,98),(52,88,99),(53,89,100),(54,90,101),(55,91,102),(56,92,103),(57,93,104),(58,94,105),(59,95,106),(60,96,107),(61,81,108),(62,82,109),(63,83,110),(64,84,111)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47),(49,57),(51,59),(53,61),(55,63),(65,73),(67,75),(69,77),(71,79),(81,89),(83,91),(85,93),(87,95),(98,106),(100,108),(102,110),(104,112),(113,121),(115,123),(117,125),(119,127),(129,137),(131,139),(133,141),(135,143)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 4A | 4B | 4C | 6A | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 8C | 8D | 8E | 8F | 12A | ··· | 12P | 12Q | ··· | 12X | 16A | ··· | 16H | 24A | ··· | 24AF | 24AG | ··· | 24AV | 48A | ··· | 48BL |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 1 | ··· | 1 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 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 | 2 | 2 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | M5(2) | C3×M5(2) |
| kernel | C32×M5(2) | C3×C48 | C6×C24 | C3×M5(2) | C3×C24 | C6×C12 | C48 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 16 | 8 | 4 | 4 | 16 | 16 | 32 | 32 | 4 | 32 |
Matrix representation of C32×M5(2) ►in GL3(𝔽97) generated by
| 61 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 61 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 35 |
| 1 | 0 | 0 |
| 0 | 22 | 95 |
| 0 | 80 | 75 |
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 22 | 96 |
G:=sub<GL(3,GF(97))| [61,0,0,0,61,0,0,0,61],[61,0,0,0,35,0,0,0,35],[1,0,0,0,22,80,0,95,75],[96,0,0,0,1,22,0,0,96] >;
C32×M5(2) in GAP, Magma, Sage, TeX
C_3^2\times M_5(2)
% in TeX
G:=Group("C3^2xM5(2)"); // GroupNames label
G:=SmallGroup(288,328);
// by ID
G=gap.SmallGroup(288,328);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,252,2045,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^9>;
// generators/relations