direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C16×He3, C32⋊4C48, C48.1C32, (C3×C48)⋊C3, C2.(C8×He3), (C3×C6).4C24, (C3×C24).8C6, C6.2(C3×C24), C3.1(C3×C48), C4.2(C4×He3), C8.2(C2×He3), C24.16(C3×C6), (C8×He3).6C2, (C2×He3).5C8, (C3×C12).10C12, C12.11(C3×C12), (C4×He3).11C4, SmallGroup(432,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16×He3
G = < a,b,c,d | a16=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Smallest permutation representation of C16×He3
►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)
(1 137 37)(2 138 38)(3 139 39)(4 140 40)(5 141 41)(6 142 42)(7 143 43)(8 144 44)(9 129 45)(10 130 46)(11 131 47)(12 132 48)(13 133 33)(14 134 34)(15 135 35)(16 136 36)(17 98 121)(18 99 122)(19 100 123)(20 101 124)(21 102 125)(22 103 126)(23 104 127)(24 105 128)(25 106 113)(26 107 114)(27 108 115)(28 109 116)(29 110 117)(30 111 118)(31 112 119)(32 97 120)(49 87 66)(50 88 67)(51 89 68)(52 90 69)(53 91 70)(54 92 71)(55 93 72)(56 94 73)(57 95 74)(58 96 75)(59 81 76)(60 82 77)(61 83 78)(62 84 79)(63 85 80)(64 86 65)
(1 90 103)(2 91 104)(3 92 105)(4 93 106)(5 94 107)(6 95 108)(7 96 109)(8 81 110)(9 82 111)(10 83 112)(11 84 97)(12 85 98)(13 86 99)(14 87 100)(15 88 101)(16 89 102)(17 48 63)(18 33 64)(19 34 49)(20 35 50)(21 36 51)(22 37 52)(23 38 53)(24 39 54)(25 40 55)(26 41 56)(27 42 57)(28 43 58)(29 44 59)(30 45 60)(31 46 61)(32 47 62)(65 122 133)(66 123 134)(67 124 135)(68 125 136)(69 126 137)(70 127 138)(71 128 139)(72 113 140)(73 114 141)(74 115 142)(75 116 143)(76 117 144)(77 118 129)(78 119 130)(79 120 131)(80 121 132)
(17 63 48)(18 64 33)(19 49 34)(20 50 35)(21 51 36)(22 52 37)(23 53 38)(24 54 39)(25 55 40)(26 56 41)(27 57 42)(28 58 43)(29 59 44)(30 60 45)(31 61 46)(32 62 47)(65 122 133)(66 123 134)(67 124 135)(68 125 136)(69 126 137)(70 127 138)(71 128 139)(72 113 140)(73 114 141)(74 115 142)(75 116 143)(76 117 144)(77 118 129)(78 119 130)(79 120 131)(80 121 132)
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), (1,137,37)(2,138,38)(3,139,39)(4,140,40)(5,141,41)(6,142,42)(7,143,43)(8,144,44)(9,129,45)(10,130,46)(11,131,47)(12,132,48)(13,133,33)(14,134,34)(15,135,35)(16,136,36)(17,98,121)(18,99,122)(19,100,123)(20,101,124)(21,102,125)(22,103,126)(23,104,127)(24,105,128)(25,106,113)(26,107,114)(27,108,115)(28,109,116)(29,110,117)(30,111,118)(31,112,119)(32,97,120)(49,87,66)(50,88,67)(51,89,68)(52,90,69)(53,91,70)(54,92,71)(55,93,72)(56,94,73)(57,95,74)(58,96,75)(59,81,76)(60,82,77)(61,83,78)(62,84,79)(63,85,80)(64,86,65), (1,90,103)(2,91,104)(3,92,105)(4,93,106)(5,94,107)(6,95,108)(7,96,109)(8,81,110)(9,82,111)(10,83,112)(11,84,97)(12,85,98)(13,86,99)(14,87,100)(15,88,101)(16,89,102)(17,48,63)(18,33,64)(19,34,49)(20,35,50)(21,36,51)(22,37,52)(23,38,53)(24,39,54)(25,40,55)(26,41,56)(27,42,57)(28,43,58)(29,44,59)(30,45,60)(31,46,61)(32,47,62)(65,122,133)(66,123,134)(67,124,135)(68,125,136)(69,126,137)(70,127,138)(71,128,139)(72,113,140)(73,114,141)(74,115,142)(75,116,143)(76,117,144)(77,118,129)(78,119,130)(79,120,131)(80,121,132), (17,63,48)(18,64,33)(19,49,34)(20,50,35)(21,51,36)(22,52,37)(23,53,38)(24,54,39)(25,55,40)(26,56,41)(27,57,42)(28,58,43)(29,59,44)(30,60,45)(31,61,46)(32,62,47)(65,122,133)(66,123,134)(67,124,135)(68,125,136)(69,126,137)(70,127,138)(71,128,139)(72,113,140)(73,114,141)(74,115,142)(75,116,143)(76,117,144)(77,118,129)(78,119,130)(79,120,131)(80,121,132)>;
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), (1,137,37)(2,138,38)(3,139,39)(4,140,40)(5,141,41)(6,142,42)(7,143,43)(8,144,44)(9,129,45)(10,130,46)(11,131,47)(12,132,48)(13,133,33)(14,134,34)(15,135,35)(16,136,36)(17,98,121)(18,99,122)(19,100,123)(20,101,124)(21,102,125)(22,103,126)(23,104,127)(24,105,128)(25,106,113)(26,107,114)(27,108,115)(28,109,116)(29,110,117)(30,111,118)(31,112,119)(32,97,120)(49,87,66)(50,88,67)(51,89,68)(52,90,69)(53,91,70)(54,92,71)(55,93,72)(56,94,73)(57,95,74)(58,96,75)(59,81,76)(60,82,77)(61,83,78)(62,84,79)(63,85,80)(64,86,65), (1,90,103)(2,91,104)(3,92,105)(4,93,106)(5,94,107)(6,95,108)(7,96,109)(8,81,110)(9,82,111)(10,83,112)(11,84,97)(12,85,98)(13,86,99)(14,87,100)(15,88,101)(16,89,102)(17,48,63)(18,33,64)(19,34,49)(20,35,50)(21,36,51)(22,37,52)(23,38,53)(24,39,54)(25,40,55)(26,41,56)(27,42,57)(28,43,58)(29,44,59)(30,45,60)(31,46,61)(32,47,62)(65,122,133)(66,123,134)(67,124,135)(68,125,136)(69,126,137)(70,127,138)(71,128,139)(72,113,140)(73,114,141)(74,115,142)(75,116,143)(76,117,144)(77,118,129)(78,119,130)(79,120,131)(80,121,132), (17,63,48)(18,64,33)(19,49,34)(20,50,35)(21,51,36)(22,52,37)(23,53,38)(24,54,39)(25,55,40)(26,56,41)(27,57,42)(28,58,43)(29,59,44)(30,60,45)(31,61,46)(32,62,47)(65,122,133)(66,123,134)(67,124,135)(68,125,136)(69,126,137)(70,127,138)(71,128,139)(72,113,140)(73,114,141)(74,115,142)(75,116,143)(76,117,144)(77,118,129)(78,119,130)(79,120,131)(80,121,132) );
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)], [(1,137,37),(2,138,38),(3,139,39),(4,140,40),(5,141,41),(6,142,42),(7,143,43),(8,144,44),(9,129,45),(10,130,46),(11,131,47),(12,132,48),(13,133,33),(14,134,34),(15,135,35),(16,136,36),(17,98,121),(18,99,122),(19,100,123),(20,101,124),(21,102,125),(22,103,126),(23,104,127),(24,105,128),(25,106,113),(26,107,114),(27,108,115),(28,109,116),(29,110,117),(30,111,118),(31,112,119),(32,97,120),(49,87,66),(50,88,67),(51,89,68),(52,90,69),(53,91,70),(54,92,71),(55,93,72),(56,94,73),(57,95,74),(58,96,75),(59,81,76),(60,82,77),(61,83,78),(62,84,79),(63,85,80),(64,86,65)], [(1,90,103),(2,91,104),(3,92,105),(4,93,106),(5,94,107),(6,95,108),(7,96,109),(8,81,110),(9,82,111),(10,83,112),(11,84,97),(12,85,98),(13,86,99),(14,87,100),(15,88,101),(16,89,102),(17,48,63),(18,33,64),(19,34,49),(20,35,50),(21,36,51),(22,37,52),(23,38,53),(24,39,54),(25,40,55),(26,41,56),(27,42,57),(28,43,58),(29,44,59),(30,45,60),(31,46,61),(32,47,62),(65,122,133),(66,123,134),(67,124,135),(68,125,136),(69,126,137),(70,127,138),(71,128,139),(72,113,140),(73,114,141),(74,115,142),(75,116,143),(76,117,144),(77,118,129),(78,119,130),(79,120,131),(80,121,132)], [(17,63,48),(18,64,33),(19,49,34),(20,50,35),(21,51,36),(22,52,37),(23,53,38),(24,54,39),(25,55,40),(26,56,41),(27,57,42),(28,58,43),(29,59,44),(30,60,45),(31,61,46),(32,62,47),(65,122,133),(66,123,134),(67,124,135),(68,125,136),(69,126,137),(70,127,138),(71,128,139),(72,113,140),(73,114,141),(74,115,142),(75,116,143),(76,117,144),(77,118,129),(78,119,130),(79,120,131),(80,121,132)]])
176 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 4A | 4B | 6A | 6B | 6C | ··· | 6J | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12T | 16A | ··· | 16H | 24A | ··· | 24H | 24I | ··· | 24AN | 48A | ··· | 48P | 48Q | ··· | 48CB |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
176 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | He3 | C2×He3 | C4×He3 | C8×He3 | C16×He3 |
| kernel | C16×He3 | C8×He3 | C3×C48 | C4×He3 | C3×C24 | C2×He3 | C3×C12 | He3 | C3×C6 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 8 | 32 | 64 | 2 | 2 | 4 | 8 | 16 |
Matrix representation of C16×He3 ►in GL3(𝔽97) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 34 | 0 |
| 0 | 96 | 1 |
| 0 | 96 | 0 |
| 35 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 35 |
| 1 | 0 | 0 |
| 36 | 61 | 0 |
| 1 | 0 | 35 |
G:=sub<GL(3,GF(97))| [8,0,0,0,8,0,0,0,8],[1,0,0,34,96,96,0,1,0],[35,0,0,0,35,0,0,0,35],[1,36,1,0,61,0,0,0,35] >;
C16×He3 in GAP, Magma, Sage, TeX
C_{16}\times {\rm He}_3 % in TeX
G:=Group("C16xHe3"); // GroupNames label
G:=SmallGroup(432,35);
// by ID
G=gap.SmallGroup(432,35);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,-3,-2,-2,126,450,192,124]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations
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