direct product, metabelian, nilpotent (class 2), monomial
Aliases: M4(2)×C3×C6, C8⋊4C62, (C6×C24)⋊22C2, (C2×C24)⋊14C6, C24⋊14(C2×C6), C4.12(C6×C12), (C6×C12).35C4, (C2×C62).9C4, (C2×C12).25C12, C12.58(C2×C12), (C3×C24)⋊34C22, C23.4(C3×C12), C4.11(C2×C62), (C2×C4).33C62, (C22×C12).37C6, C12.67(C22×C6), C6.41(C22×C12), (C22×C6).13C12, C62.120(C2×C4), C22.11(C6×C12), (C6×C12).382C22, (C3×C12).194C23, (C2×C8)⋊6(C3×C6), C2.6(C2×C6×C12), (C2×C6×C12).26C2, (C2×C4).6(C3×C12), (C2×C6).33(C2×C12), (C22×C4).8(C3×C6), (C3×C12).144(C2×C4), (C2×C12).174(C2×C6), (C3×C6).133(C22×C4), SmallGroup(288,827)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C3×C6
G = < a,b,c,d | a3=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >
Subgroups: 228 in 204 conjugacy classes, 180 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C32, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3×C6, C3×C6, C3×C6, C24, C2×C12, C22×C6, C2×M4(2), C3×C12, C3×C12, C62, C62, C62, C2×C24, C3×M4(2), C22×C12, C3×C24, C6×C12, C6×C12, C2×C62, C6×M4(2), C6×C24, C32×M4(2), C2×C6×C12, M4(2)×C3×C6
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, M4(2), C22×C4, C3×C6, C2×C12, C22×C6, C2×M4(2), C3×C12, C62, C3×M4(2), C22×C12, C6×C12, C2×C62, C6×M4(2), C32×M4(2), C2×C6×C12, M4(2)×C3×C6
►On 144 points
Generators in S144
(1 103 95)(2 104 96)(3 97 89)(4 98 90)(5 99 91)(6 100 92)(7 101 93)(8 102 94)(9 106 134)(10 107 135)(11 108 136)(12 109 129)(13 110 130)(14 111 131)(15 112 132)(16 105 133)(17 125 117)(18 126 118)(19 127 119)(20 128 120)(21 121 113)(22 122 114)(23 123 115)(24 124 116)(25 53 45)(26 54 46)(27 55 47)(28 56 48)(29 49 41)(30 50 42)(31 51 43)(32 52 44)(33 141 81)(34 142 82)(35 143 83)(36 144 84)(37 137 85)(38 138 86)(39 139 87)(40 140 88)(57 76 65)(58 77 66)(59 78 67)(60 79 68)(61 80 69)(62 73 70)(63 74 71)(64 75 72)
(1 35 23 63 43 135)(2 36 24 64 44 136)(3 37 17 57 45 129)(4 38 18 58 46 130)(5 39 19 59 47 131)(6 40 20 60 48 132)(7 33 21 61 41 133)(8 34 22 62 42 134)(9 102 142 122 73 30)(10 103 143 123 74 31)(11 104 144 124 75 32)(12 97 137 125 76 25)(13 98 138 126 77 26)(14 99 139 127 78 27)(15 100 140 128 79 28)(16 101 141 121 80 29)(49 105 93 81 113 69)(50 106 94 82 114 70)(51 107 95 83 115 71)(52 108 96 84 116 72)(53 109 89 85 117 65)(54 110 90 86 118 66)(55 111 91 87 119 67)(56 112 92 88 120 68)
(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 59)(2 64)(3 61)(4 58)(5 63)(6 60)(7 57)(8 62)(9 122)(10 127)(11 124)(12 121)(13 126)(14 123)(15 128)(16 125)(17 133)(18 130)(19 135)(20 132)(21 129)(22 134)(23 131)(24 136)(25 141)(26 138)(27 143)(28 140)(29 137)(30 142)(31 139)(32 144)(33 45)(34 42)(35 47)(36 44)(37 41)(38 46)(39 43)(40 48)(49 85)(50 82)(51 87)(52 84)(53 81)(54 86)(55 83)(56 88)(65 93)(66 90)(67 95)(68 92)(69 89)(70 94)(71 91)(72 96)(73 102)(74 99)(75 104)(76 101)(77 98)(78 103)(79 100)(80 97)(105 117)(106 114)(107 119)(108 116)(109 113)(110 118)(111 115)(112 120)
G:=sub<Sym(144)| (1,103,95)(2,104,96)(3,97,89)(4,98,90)(5,99,91)(6,100,92)(7,101,93)(8,102,94)(9,106,134)(10,107,135)(11,108,136)(12,109,129)(13,110,130)(14,111,131)(15,112,132)(16,105,133)(17,125,117)(18,126,118)(19,127,119)(20,128,120)(21,121,113)(22,122,114)(23,123,115)(24,124,116)(25,53,45)(26,54,46)(27,55,47)(28,56,48)(29,49,41)(30,50,42)(31,51,43)(32,52,44)(33,141,81)(34,142,82)(35,143,83)(36,144,84)(37,137,85)(38,138,86)(39,139,87)(40,140,88)(57,76,65)(58,77,66)(59,78,67)(60,79,68)(61,80,69)(62,73,70)(63,74,71)(64,75,72), (1,35,23,63,43,135)(2,36,24,64,44,136)(3,37,17,57,45,129)(4,38,18,58,46,130)(5,39,19,59,47,131)(6,40,20,60,48,132)(7,33,21,61,41,133)(8,34,22,62,42,134)(9,102,142,122,73,30)(10,103,143,123,74,31)(11,104,144,124,75,32)(12,97,137,125,76,25)(13,98,138,126,77,26)(14,99,139,127,78,27)(15,100,140,128,79,28)(16,101,141,121,80,29)(49,105,93,81,113,69)(50,106,94,82,114,70)(51,107,95,83,115,71)(52,108,96,84,116,72)(53,109,89,85,117,65)(54,110,90,86,118,66)(55,111,91,87,119,67)(56,112,92,88,120,68), (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), (1,59)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,122)(10,127)(11,124)(12,121)(13,126)(14,123)(15,128)(16,125)(17,133)(18,130)(19,135)(20,132)(21,129)(22,134)(23,131)(24,136)(25,141)(26,138)(27,143)(28,140)(29,137)(30,142)(31,139)(32,144)(33,45)(34,42)(35,47)(36,44)(37,41)(38,46)(39,43)(40,48)(49,85)(50,82)(51,87)(52,84)(53,81)(54,86)(55,83)(56,88)(65,93)(66,90)(67,95)(68,92)(69,89)(70,94)(71,91)(72,96)(73,102)(74,99)(75,104)(76,101)(77,98)(78,103)(79,100)(80,97)(105,117)(106,114)(107,119)(108,116)(109,113)(110,118)(111,115)(112,120)>;
G:=Group( (1,103,95)(2,104,96)(3,97,89)(4,98,90)(5,99,91)(6,100,92)(7,101,93)(8,102,94)(9,106,134)(10,107,135)(11,108,136)(12,109,129)(13,110,130)(14,111,131)(15,112,132)(16,105,133)(17,125,117)(18,126,118)(19,127,119)(20,128,120)(21,121,113)(22,122,114)(23,123,115)(24,124,116)(25,53,45)(26,54,46)(27,55,47)(28,56,48)(29,49,41)(30,50,42)(31,51,43)(32,52,44)(33,141,81)(34,142,82)(35,143,83)(36,144,84)(37,137,85)(38,138,86)(39,139,87)(40,140,88)(57,76,65)(58,77,66)(59,78,67)(60,79,68)(61,80,69)(62,73,70)(63,74,71)(64,75,72), (1,35,23,63,43,135)(2,36,24,64,44,136)(3,37,17,57,45,129)(4,38,18,58,46,130)(5,39,19,59,47,131)(6,40,20,60,48,132)(7,33,21,61,41,133)(8,34,22,62,42,134)(9,102,142,122,73,30)(10,103,143,123,74,31)(11,104,144,124,75,32)(12,97,137,125,76,25)(13,98,138,126,77,26)(14,99,139,127,78,27)(15,100,140,128,79,28)(16,101,141,121,80,29)(49,105,93,81,113,69)(50,106,94,82,114,70)(51,107,95,83,115,71)(52,108,96,84,116,72)(53,109,89,85,117,65)(54,110,90,86,118,66)(55,111,91,87,119,67)(56,112,92,88,120,68), (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), (1,59)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,122)(10,127)(11,124)(12,121)(13,126)(14,123)(15,128)(16,125)(17,133)(18,130)(19,135)(20,132)(21,129)(22,134)(23,131)(24,136)(25,141)(26,138)(27,143)(28,140)(29,137)(30,142)(31,139)(32,144)(33,45)(34,42)(35,47)(36,44)(37,41)(38,46)(39,43)(40,48)(49,85)(50,82)(51,87)(52,84)(53,81)(54,86)(55,83)(56,88)(65,93)(66,90)(67,95)(68,92)(69,89)(70,94)(71,91)(72,96)(73,102)(74,99)(75,104)(76,101)(77,98)(78,103)(79,100)(80,97)(105,117)(106,114)(107,119)(108,116)(109,113)(110,118)(111,115)(112,120) );
G=PermutationGroup([[(1,103,95),(2,104,96),(3,97,89),(4,98,90),(5,99,91),(6,100,92),(7,101,93),(8,102,94),(9,106,134),(10,107,135),(11,108,136),(12,109,129),(13,110,130),(14,111,131),(15,112,132),(16,105,133),(17,125,117),(18,126,118),(19,127,119),(20,128,120),(21,121,113),(22,122,114),(23,123,115),(24,124,116),(25,53,45),(26,54,46),(27,55,47),(28,56,48),(29,49,41),(30,50,42),(31,51,43),(32,52,44),(33,141,81),(34,142,82),(35,143,83),(36,144,84),(37,137,85),(38,138,86),(39,139,87),(40,140,88),(57,76,65),(58,77,66),(59,78,67),(60,79,68),(61,80,69),(62,73,70),(63,74,71),(64,75,72)], [(1,35,23,63,43,135),(2,36,24,64,44,136),(3,37,17,57,45,129),(4,38,18,58,46,130),(5,39,19,59,47,131),(6,40,20,60,48,132),(7,33,21,61,41,133),(8,34,22,62,42,134),(9,102,142,122,73,30),(10,103,143,123,74,31),(11,104,144,124,75,32),(12,97,137,125,76,25),(13,98,138,126,77,26),(14,99,139,127,78,27),(15,100,140,128,79,28),(16,101,141,121,80,29),(49,105,93,81,113,69),(50,106,94,82,114,70),(51,107,95,83,115,71),(52,108,96,84,116,72),(53,109,89,85,117,65),(54,110,90,86,118,66),(55,111,91,87,119,67),(56,112,92,88,120,68)], [(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)], [(1,59),(2,64),(3,61),(4,58),(5,63),(6,60),(7,57),(8,62),(9,122),(10,127),(11,124),(12,121),(13,126),(14,123),(15,128),(16,125),(17,133),(18,130),(19,135),(20,132),(21,129),(22,134),(23,131),(24,136),(25,141),(26,138),(27,143),(28,140),(29,137),(30,142),(31,139),(32,144),(33,45),(34,42),(35,47),(36,44),(37,41),(38,46),(39,43),(40,48),(49,85),(50,82),(51,87),(52,84),(53,81),(54,86),(55,83),(56,88),(65,93),(66,90),(67,95),(68,92),(69,89),(70,94),(71,91),(72,96),(73,102),(74,99),(75,104),(76,101),(77,98),(78,103),(79,100),(80,97),(105,117),(106,114),(107,119),(108,116),(109,113),(110,118),(111,115),(112,120)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6X | 6Y | ··· | 6AN | 8A | ··· | 8H | 12A | ··· | 12AF | 12AG | ··· | 12AV | 24A | ··· | 24BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 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 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) |
| kernel | M4(2)×C3×C6 | C6×C24 | C32×M4(2) | C2×C6×C12 | C6×M4(2) | C6×C12 | C2×C62 | C2×C24 | C3×M4(2) | C22×C12 | C2×C12 | C22×C6 | C3×C6 | C6 |
| # reps | 1 | 2 | 4 | 1 | 8 | 6 | 2 | 16 | 32 | 8 | 48 | 16 | 4 | 32 |
Matrix representation of M4(2)×C3×C6 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 9 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 72 | 0 | 0 |
| 0 | 54 | 71 |
| 0 | 48 | 19 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 19 | 1 |
G:=sub<GL(3,GF(73))| [8,0,0,0,64,0,0,0,64],[9,0,0,0,1,0,0,0,1],[72,0,0,0,54,48,0,71,19],[1,0,0,0,72,19,0,0,1] >;
M4(2)×C3×C6 in GAP, Magma, Sage, TeX
M_4(2)\times C_3\times C_6
% in TeX
G:=Group("M4(2)xC3xC6"); // GroupNames label
G:=SmallGroup(288,827);
// by ID
G=gap.SmallGroup(288,827);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,2045,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^8=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^5>;
// generators/relations