direct product, metabelian, nilpotent (class 3), monomial
Aliases: C32×C4.10D4, (C2×C12).4C12, (C6×C12).14C4, C12.76(C3×D4), (C2×C4).2C62, (C6×Q8).22C6, (C3×C12).177D4, C22.4(C6×C12), C62.90(C2×C4), C4.10(D4×C32), (C3×M4(2)).9C6, M4(2).1(C3×C6), (C6×C12).259C22, (C32×M4(2)).5C2, (C2×C4).(C3×C12), (Q8×C3×C6).11C2, (C2×Q8).3(C3×C6), (C2×C6).31(C2×C12), (C2×C12).68(C2×C6), C6.32(C3×C22⋊C4), C2.5(C32×C22⋊C4), (C3×C6).81(C22⋊C4), SmallGroup(288,319)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4.10D4
G = < a,b,c,d,e | a3=b3=c4=1, d4=c2, e2=dcd-1=c-1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >
Subgroups: 156 in 114 conjugacy classes, 72 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, C12, C12, C2×C6, M4(2), C2×Q8, C3×C6, C3×C6, C24, C2×C12, C3×Q8, C4.10D4, C3×C12, C3×C12, C62, C3×M4(2), C6×Q8, C3×C24, C6×C12, C6×C12, Q8×C32, C3×C4.10D4, C32×M4(2), Q8×C3×C6, C32×C4.10D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C32, C12, C2×C6, C22⋊C4, C3×C6, C2×C12, C3×D4, C4.10D4, C3×C12, C62, C3×C22⋊C4, C6×C12, D4×C32, C3×C4.10D4, C32×C22⋊C4, C32×C4.10D4
►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)(9 106 130)(10 107 131)(11 108 132)(12 109 133)(13 110 134)(14 111 135)(15 112 136)(16 105 129)(17 121 113)(18 122 114)(19 123 115)(20 124 116)(21 125 117)(22 126 118)(23 127 119)(24 128 120)(25 49 41)(26 50 42)(27 51 43)(28 52 44)(29 53 45)(30 54 46)(31 55 47)(32 56 48)(33 137 81)(34 138 82)(35 139 83)(36 140 84)(37 141 85)(38 142 86)(39 143 87)(40 144 88)(57 77 65)(58 78 66)(59 79 67)(60 80 68)(61 73 69)(62 74 70)(63 75 71)(64 76 72)
(1 23 47)(2 24 48)(3 17 41)(4 18 42)(5 19 43)(6 20 44)(7 21 45)(8 22 46)(9 138 78)(10 139 79)(11 140 80)(12 141 73)(13 142 74)(14 143 75)(15 144 76)(16 137 77)(25 97 121)(26 98 122)(27 99 123)(28 100 124)(29 101 125)(30 102 126)(31 103 127)(32 104 128)(33 57 129)(34 58 130)(35 59 131)(36 60 132)(37 61 133)(38 62 134)(39 63 135)(40 64 136)(49 89 113)(50 90 114)(51 91 115)(52 92 116)(53 93 117)(54 94 118)(55 95 119)(56 96 120)(65 105 81)(66 106 82)(67 107 83)(68 108 84)(69 109 85)(70 110 86)(71 111 87)(72 112 88)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 75 77 79)(74 80 78 76)(81 83 85 87)(82 88 86 84)(89 91 93 95)(90 96 94 92)(97 99 101 103)(98 104 102 100)(105 107 109 111)(106 112 110 108)(113 115 117 119)(114 120 118 116)(121 123 125 127)(122 128 126 124)(129 131 133 135)(130 136 134 132)(137 139 141 143)(138 144 142 140)
(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 60 7 62 5 64 3 58)(2 61 4 59 6 57 8 63)(9 127 11 125 13 123 15 121)(10 124 16 126 14 128 12 122)(17 130 23 132 21 134 19 136)(18 131 20 129 22 135 24 133)(25 138 31 140 29 142 27 144)(26 139 28 137 30 143 32 141)(33 46 39 48 37 42 35 44)(34 47 36 45 38 43 40 41)(49 82 55 84 53 86 51 88)(50 83 52 81 54 87 56 85)(65 94 71 96 69 90 67 92)(66 95 68 93 70 91 72 89)(73 98 79 100 77 102 75 104)(74 99 76 97 78 103 80 101)(105 118 111 120 109 114 107 116)(106 119 108 117 110 115 112 113)
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)(9,106,130)(10,107,131)(11,108,132)(12,109,133)(13,110,134)(14,111,135)(15,112,136)(16,105,129)(17,121,113)(18,122,114)(19,123,115)(20,124,116)(21,125,117)(22,126,118)(23,127,119)(24,128,120)(25,49,41)(26,50,42)(27,51,43)(28,52,44)(29,53,45)(30,54,46)(31,55,47)(32,56,48)(33,137,81)(34,138,82)(35,139,83)(36,140,84)(37,141,85)(38,142,86)(39,143,87)(40,144,88)(57,77,65)(58,78,66)(59,79,67)(60,80,68)(61,73,69)(62,74,70)(63,75,71)(64,76,72), (1,23,47)(2,24,48)(3,17,41)(4,18,42)(5,19,43)(6,20,44)(7,21,45)(8,22,46)(9,138,78)(10,139,79)(11,140,80)(12,141,73)(13,142,74)(14,143,75)(15,144,76)(16,137,77)(25,97,121)(26,98,122)(27,99,123)(28,100,124)(29,101,125)(30,102,126)(31,103,127)(32,104,128)(33,57,129)(34,58,130)(35,59,131)(36,60,132)(37,61,133)(38,62,134)(39,63,135)(40,64,136)(49,89,113)(50,90,114)(51,91,115)(52,92,116)(53,93,117)(54,94,118)(55,95,119)(56,96,120)(65,105,81)(66,106,82)(67,107,83)(68,108,84)(69,109,85)(70,110,86)(71,111,87)(72,112,88), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,75,77,79)(74,80,78,76)(81,83,85,87)(82,88,86,84)(89,91,93,95)(90,96,94,92)(97,99,101,103)(98,104,102,100)(105,107,109,111)(106,112,110,108)(113,115,117,119)(114,120,118,116)(121,123,125,127)(122,128,126,124)(129,131,133,135)(130,136,134,132)(137,139,141,143)(138,144,142,140), (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), (1,60,7,62,5,64,3,58)(2,61,4,59,6,57,8,63)(9,127,11,125,13,123,15,121)(10,124,16,126,14,128,12,122)(17,130,23,132,21,134,19,136)(18,131,20,129,22,135,24,133)(25,138,31,140,29,142,27,144)(26,139,28,137,30,143,32,141)(33,46,39,48,37,42,35,44)(34,47,36,45,38,43,40,41)(49,82,55,84,53,86,51,88)(50,83,52,81,54,87,56,85)(65,94,71,96,69,90,67,92)(66,95,68,93,70,91,72,89)(73,98,79,100,77,102,75,104)(74,99,76,97,78,103,80,101)(105,118,111,120,109,114,107,116)(106,119,108,117,110,115,112,113)>;
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)(9,106,130)(10,107,131)(11,108,132)(12,109,133)(13,110,134)(14,111,135)(15,112,136)(16,105,129)(17,121,113)(18,122,114)(19,123,115)(20,124,116)(21,125,117)(22,126,118)(23,127,119)(24,128,120)(25,49,41)(26,50,42)(27,51,43)(28,52,44)(29,53,45)(30,54,46)(31,55,47)(32,56,48)(33,137,81)(34,138,82)(35,139,83)(36,140,84)(37,141,85)(38,142,86)(39,143,87)(40,144,88)(57,77,65)(58,78,66)(59,79,67)(60,80,68)(61,73,69)(62,74,70)(63,75,71)(64,76,72), (1,23,47)(2,24,48)(3,17,41)(4,18,42)(5,19,43)(6,20,44)(7,21,45)(8,22,46)(9,138,78)(10,139,79)(11,140,80)(12,141,73)(13,142,74)(14,143,75)(15,144,76)(16,137,77)(25,97,121)(26,98,122)(27,99,123)(28,100,124)(29,101,125)(30,102,126)(31,103,127)(32,104,128)(33,57,129)(34,58,130)(35,59,131)(36,60,132)(37,61,133)(38,62,134)(39,63,135)(40,64,136)(49,89,113)(50,90,114)(51,91,115)(52,92,116)(53,93,117)(54,94,118)(55,95,119)(56,96,120)(65,105,81)(66,106,82)(67,107,83)(68,108,84)(69,109,85)(70,110,86)(71,111,87)(72,112,88), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,75,77,79)(74,80,78,76)(81,83,85,87)(82,88,86,84)(89,91,93,95)(90,96,94,92)(97,99,101,103)(98,104,102,100)(105,107,109,111)(106,112,110,108)(113,115,117,119)(114,120,118,116)(121,123,125,127)(122,128,126,124)(129,131,133,135)(130,136,134,132)(137,139,141,143)(138,144,142,140), (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), (1,60,7,62,5,64,3,58)(2,61,4,59,6,57,8,63)(9,127,11,125,13,123,15,121)(10,124,16,126,14,128,12,122)(17,130,23,132,21,134,19,136)(18,131,20,129,22,135,24,133)(25,138,31,140,29,142,27,144)(26,139,28,137,30,143,32,141)(33,46,39,48,37,42,35,44)(34,47,36,45,38,43,40,41)(49,82,55,84,53,86,51,88)(50,83,52,81,54,87,56,85)(65,94,71,96,69,90,67,92)(66,95,68,93,70,91,72,89)(73,98,79,100,77,102,75,104)(74,99,76,97,78,103,80,101)(105,118,111,120,109,114,107,116)(106,119,108,117,110,115,112,113) );
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),(9,106,130),(10,107,131),(11,108,132),(12,109,133),(13,110,134),(14,111,135),(15,112,136),(16,105,129),(17,121,113),(18,122,114),(19,123,115),(20,124,116),(21,125,117),(22,126,118),(23,127,119),(24,128,120),(25,49,41),(26,50,42),(27,51,43),(28,52,44),(29,53,45),(30,54,46),(31,55,47),(32,56,48),(33,137,81),(34,138,82),(35,139,83),(36,140,84),(37,141,85),(38,142,86),(39,143,87),(40,144,88),(57,77,65),(58,78,66),(59,79,67),(60,80,68),(61,73,69),(62,74,70),(63,75,71),(64,76,72)], [(1,23,47),(2,24,48),(3,17,41),(4,18,42),(5,19,43),(6,20,44),(7,21,45),(8,22,46),(9,138,78),(10,139,79),(11,140,80),(12,141,73),(13,142,74),(14,143,75),(15,144,76),(16,137,77),(25,97,121),(26,98,122),(27,99,123),(28,100,124),(29,101,125),(30,102,126),(31,103,127),(32,104,128),(33,57,129),(34,58,130),(35,59,131),(36,60,132),(37,61,133),(38,62,134),(39,63,135),(40,64,136),(49,89,113),(50,90,114),(51,91,115),(52,92,116),(53,93,117),(54,94,118),(55,95,119),(56,96,120),(65,105,81),(66,106,82),(67,107,83),(68,108,84),(69,109,85),(70,110,86),(71,111,87),(72,112,88)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,19,21,23),(18,24,22,20),(25,27,29,31),(26,32,30,28),(33,35,37,39),(34,40,38,36),(41,43,45,47),(42,48,46,44),(49,51,53,55),(50,56,54,52),(57,59,61,63),(58,64,62,60),(65,67,69,71),(66,72,70,68),(73,75,77,79),(74,80,78,76),(81,83,85,87),(82,88,86,84),(89,91,93,95),(90,96,94,92),(97,99,101,103),(98,104,102,100),(105,107,109,111),(106,112,110,108),(113,115,117,119),(114,120,118,116),(121,123,125,127),(122,128,126,124),(129,131,133,135),(130,136,134,132),(137,139,141,143),(138,144,142,140)], [(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)], [(1,60,7,62,5,64,3,58),(2,61,4,59,6,57,8,63),(9,127,11,125,13,123,15,121),(10,124,16,126,14,128,12,122),(17,130,23,132,21,134,19,136),(18,131,20,129,22,135,24,133),(25,138,31,140,29,142,27,144),(26,139,28,137,30,143,32,141),(33,46,39,48,37,42,35,44),(34,47,36,45,38,43,40,41),(49,82,55,84,53,86,51,88),(50,83,52,81,54,87,56,85),(65,94,71,96,69,90,67,92),(66,95,68,93,70,91,72,89),(73,98,79,100,77,102,75,104),(74,99,76,97,78,103,80,101),(105,118,111,120,109,114,107,116),(106,119,108,117,110,115,112,113)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12AF | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | C3×D4 | C4.10D4 | C3×C4.10D4 |
| kernel | C32×C4.10D4 | C32×M4(2) | Q8×C3×C6 | C3×C4.10D4 | C6×C12 | C3×M4(2) | C6×Q8 | C2×C12 | C3×C12 | C12 | C32 | C3 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 | 2 | 16 | 1 | 8 |
Matrix representation of C32×C4.10D4 ►in GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 72 |
| 0 | 0 | 0 | 46 | 1 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 71 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 27 | 0 |
| 0 | 0 | 1 | 0 | 46 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 12 | 61 |
| 0 | 0 | 57 | 0 | 0 | 12 |
| 0 | 0 | 6 | 67 | 57 | 16 |
| 0 | 0 | 0 | 6 | 16 | 57 |
G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,27,0,0,0,71,1,27,46,0,0,0,0,0,1,0,0,0,0,72,0],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,46,0,0,1,0,0,0,0,1,0,0,0,71,1,27,46,0,0,0,1,0,0],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,32,57,6,0,0,0,0,0,67,6,0,0,12,0,57,16,0,0,61,12,16,57] >;
C32×C4.10D4 in GAP, Magma, Sage, TeX
C_3^2\times C_4._{10}D_4 % in TeX
G:=Group("C3^2xC4.10D4"); // GroupNames label
G:=SmallGroup(288,319);
// by ID
G=gap.SmallGroup(288,319);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,1016,6304,4548,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=1,d^4=c^2,e^2=d*c*d^-1=c^-1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations