Aliases: He3⋊2C16, C4.2(He3⋊C4), C2.(He3⋊2C8), (C2×He3).2C8, (C4×He3).1C4, C12.7(C32⋊C4), C3.(C32⋊2C16), He3⋊4C8.1C2, C6.2(C32⋊2C8), SmallGroup(432,57)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3⋊2C16 |
Generators and relations for He3⋊2C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, cac-1=ab-1, dad-1=abc, bc=cb, bd=db, dcd-1=ac-1 >
Smallest permutation representation of He3⋊2C16
►On 144 points
Generators in S144
(2 118 67)(3 97 58)(4 95 32)(6 122 71)(7 101 62)(8 83 20)(10 126 75)(11 105 50)(12 87 24)(14 114 79)(15 109 54)(16 91 28)(17 139 121)(18 38 61)(19 141 123)(21 143 125)(22 42 49)(23 129 127)(25 131 113)(26 46 53)(27 133 115)(29 135 117)(30 34 57)(31 137 119)(33 66 92)(36 120 98)(37 70 96)(40 124 102)(41 74 84)(44 128 106)(45 78 88)(48 116 110)(51 77 130)(55 65 134)(59 69 138)(63 73 142)(81 100 140)(85 104 144)(89 108 132)(93 112 136)
(1 111 56)(2 112 57)(3 97 58)(4 98 59)(5 99 60)(6 100 61)(7 101 62)(8 102 63)(9 103 64)(10 104 49)(11 105 50)(12 106 51)(13 107 52)(14 108 53)(15 109 54)(16 110 55)(17 121 139)(18 122 140)(19 123 141)(20 124 142)(21 125 143)(22 126 144)(23 127 129)(24 128 130)(25 113 131)(26 114 132)(27 115 133)(28 116 134)(29 117 135)(30 118 136)(31 119 137)(32 120 138)(33 66 92)(34 67 93)(35 68 94)(36 69 95)(37 70 96)(38 71 81)(39 72 82)(40 73 83)(41 74 84)(42 75 85)(43 76 86)(44 77 87)(45 78 88)(46 79 89)(47 80 90)(48 65 91)
(1 29 33)(2 34 118)(3 35 31)(4 120 36)(5 17 37)(6 38 122)(7 39 19)(8 124 40)(9 21 41)(10 42 126)(11 43 23)(12 128 44)(13 25 45)(14 46 114)(15 47 27)(16 116 48)(18 61 81)(20 83 63)(22 49 85)(24 87 51)(26 53 89)(28 91 55)(30 57 93)(32 95 59)(50 86 129)(52 131 88)(54 90 133)(56 135 92)(58 94 137)(60 139 96)(62 82 141)(64 143 84)(65 110 134)(66 111 117)(67 136 112)(68 119 97)(69 98 138)(70 99 121)(71 140 100)(72 123 101)(73 102 142)(74 103 125)(75 144 104)(76 127 105)(77 106 130)(78 107 113)(79 132 108)(80 115 109)
(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)
G:=sub<Sym(144)| (2,118,67)(3,97,58)(4,95,32)(6,122,71)(7,101,62)(8,83,20)(10,126,75)(11,105,50)(12,87,24)(14,114,79)(15,109,54)(16,91,28)(17,139,121)(18,38,61)(19,141,123)(21,143,125)(22,42,49)(23,129,127)(25,131,113)(26,46,53)(27,133,115)(29,135,117)(30,34,57)(31,137,119)(33,66,92)(36,120,98)(37,70,96)(40,124,102)(41,74,84)(44,128,106)(45,78,88)(48,116,110)(51,77,130)(55,65,134)(59,69,138)(63,73,142)(81,100,140)(85,104,144)(89,108,132)(93,112,136), (1,111,56)(2,112,57)(3,97,58)(4,98,59)(5,99,60)(6,100,61)(7,101,62)(8,102,63)(9,103,64)(10,104,49)(11,105,50)(12,106,51)(13,107,52)(14,108,53)(15,109,54)(16,110,55)(17,121,139)(18,122,140)(19,123,141)(20,124,142)(21,125,143)(22,126,144)(23,127,129)(24,128,130)(25,113,131)(26,114,132)(27,115,133)(28,116,134)(29,117,135)(30,118,136)(31,119,137)(32,120,138)(33,66,92)(34,67,93)(35,68,94)(36,69,95)(37,70,96)(38,71,81)(39,72,82)(40,73,83)(41,74,84)(42,75,85)(43,76,86)(44,77,87)(45,78,88)(46,79,89)(47,80,90)(48,65,91), (1,29,33)(2,34,118)(3,35,31)(4,120,36)(5,17,37)(6,38,122)(7,39,19)(8,124,40)(9,21,41)(10,42,126)(11,43,23)(12,128,44)(13,25,45)(14,46,114)(15,47,27)(16,116,48)(18,61,81)(20,83,63)(22,49,85)(24,87,51)(26,53,89)(28,91,55)(30,57,93)(32,95,59)(50,86,129)(52,131,88)(54,90,133)(56,135,92)(58,94,137)(60,139,96)(62,82,141)(64,143,84)(65,110,134)(66,111,117)(67,136,112)(68,119,97)(69,98,138)(70,99,121)(71,140,100)(72,123,101)(73,102,142)(74,103,125)(75,144,104)(76,127,105)(77,106,130)(78,107,113)(79,132,108)(80,115,109), (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)>;
G:=Group( (2,118,67)(3,97,58)(4,95,32)(6,122,71)(7,101,62)(8,83,20)(10,126,75)(11,105,50)(12,87,24)(14,114,79)(15,109,54)(16,91,28)(17,139,121)(18,38,61)(19,141,123)(21,143,125)(22,42,49)(23,129,127)(25,131,113)(26,46,53)(27,133,115)(29,135,117)(30,34,57)(31,137,119)(33,66,92)(36,120,98)(37,70,96)(40,124,102)(41,74,84)(44,128,106)(45,78,88)(48,116,110)(51,77,130)(55,65,134)(59,69,138)(63,73,142)(81,100,140)(85,104,144)(89,108,132)(93,112,136), (1,111,56)(2,112,57)(3,97,58)(4,98,59)(5,99,60)(6,100,61)(7,101,62)(8,102,63)(9,103,64)(10,104,49)(11,105,50)(12,106,51)(13,107,52)(14,108,53)(15,109,54)(16,110,55)(17,121,139)(18,122,140)(19,123,141)(20,124,142)(21,125,143)(22,126,144)(23,127,129)(24,128,130)(25,113,131)(26,114,132)(27,115,133)(28,116,134)(29,117,135)(30,118,136)(31,119,137)(32,120,138)(33,66,92)(34,67,93)(35,68,94)(36,69,95)(37,70,96)(38,71,81)(39,72,82)(40,73,83)(41,74,84)(42,75,85)(43,76,86)(44,77,87)(45,78,88)(46,79,89)(47,80,90)(48,65,91), (1,29,33)(2,34,118)(3,35,31)(4,120,36)(5,17,37)(6,38,122)(7,39,19)(8,124,40)(9,21,41)(10,42,126)(11,43,23)(12,128,44)(13,25,45)(14,46,114)(15,47,27)(16,116,48)(18,61,81)(20,83,63)(22,49,85)(24,87,51)(26,53,89)(28,91,55)(30,57,93)(32,95,59)(50,86,129)(52,131,88)(54,90,133)(56,135,92)(58,94,137)(60,139,96)(62,82,141)(64,143,84)(65,110,134)(66,111,117)(67,136,112)(68,119,97)(69,98,138)(70,99,121)(71,140,100)(72,123,101)(73,102,142)(74,103,125)(75,144,104)(76,127,105)(77,106,130)(78,107,113)(79,132,108)(80,115,109), (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) );
G=PermutationGroup([[(2,118,67),(3,97,58),(4,95,32),(6,122,71),(7,101,62),(8,83,20),(10,126,75),(11,105,50),(12,87,24),(14,114,79),(15,109,54),(16,91,28),(17,139,121),(18,38,61),(19,141,123),(21,143,125),(22,42,49),(23,129,127),(25,131,113),(26,46,53),(27,133,115),(29,135,117),(30,34,57),(31,137,119),(33,66,92),(36,120,98),(37,70,96),(40,124,102),(41,74,84),(44,128,106),(45,78,88),(48,116,110),(51,77,130),(55,65,134),(59,69,138),(63,73,142),(81,100,140),(85,104,144),(89,108,132),(93,112,136)], [(1,111,56),(2,112,57),(3,97,58),(4,98,59),(5,99,60),(6,100,61),(7,101,62),(8,102,63),(9,103,64),(10,104,49),(11,105,50),(12,106,51),(13,107,52),(14,108,53),(15,109,54),(16,110,55),(17,121,139),(18,122,140),(19,123,141),(20,124,142),(21,125,143),(22,126,144),(23,127,129),(24,128,130),(25,113,131),(26,114,132),(27,115,133),(28,116,134),(29,117,135),(30,118,136),(31,119,137),(32,120,138),(33,66,92),(34,67,93),(35,68,94),(36,69,95),(37,70,96),(38,71,81),(39,72,82),(40,73,83),(41,74,84),(42,75,85),(43,76,86),(44,77,87),(45,78,88),(46,79,89),(47,80,90),(48,65,91)], [(1,29,33),(2,34,118),(3,35,31),(4,120,36),(5,17,37),(6,38,122),(7,39,19),(8,124,40),(9,21,41),(10,42,126),(11,43,23),(12,128,44),(13,25,45),(14,46,114),(15,47,27),(16,116,48),(18,61,81),(20,83,63),(22,49,85),(24,87,51),(26,53,89),(28,91,55),(30,57,93),(32,95,59),(50,86,129),(52,131,88),(54,90,133),(56,135,92),(58,94,137),(60,139,96),(62,82,141),(64,143,84),(65,110,134),(66,111,117),(67,136,112),(68,119,97),(69,98,138),(70,99,121),(71,140,100),(72,123,101),(73,102,142),(74,103,125),(75,144,104),(76,127,105),(77,106,130),(78,107,113),(79,132,108),(80,115,109)], [(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)]])
56 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 1 | 1 | 1 | 1 | 12 | 12 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | - | |||||||
| image | C1 | C2 | C4 | C8 | C16 | He3⋊C4 | He3⋊2C8 | He3⋊2C16 | C32⋊C4 | C32⋊2C8 | C32⋊2C16 |
| kernel | He3⋊2C16 | He3⋊4C8 | C4×He3 | C2×He3 | He3 | C4 | C2 | C1 | C12 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 8 | 8 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of He3⋊2C16 ►in GL4(𝔽97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 35 |
| 1 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 61 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 77 | 76 | 76 |
| 0 | 76 | 77 | 76 |
| 0 | 77 | 77 | 41 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,61,0,0,0,0,35],[1,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[18,0,0,0,0,77,76,77,0,76,77,77,0,76,76,41] >;
He3⋊2C16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_2C_{16} % in TeX
G:=Group("He3:2C16"); // GroupNames label
G:=SmallGroup(432,57);
// by ID
G=gap.SmallGroup(432,57);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,14,36,58,3924,571,5381,5052,537]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations
Export