non-abelian, supersoluble, monomial
Aliases: He3⋊4C16, (C3×C24).8S3, C32⋊3(C3⋊C16), C2.(He3⋊4C8), (C4×He3).7C4, (C8×He3).5C2, (C2×He3).4C8, C24.14(C3⋊S3), (C3×C12).7Dic3, C4.2(He3⋊3C4), C8.2(He3⋊C2), C6.4(C32⋊4C8), C3.2(C24.S3), C12.12(C3⋊Dic3), (C3×C6).4(C3⋊C8), SmallGroup(432,33)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3⋊4C16 |
Generators and relations for He3⋊4C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of He3⋊4C16
►On 144 points
Generators in S144
(17 55 128)(18 113 56)(19 57 114)(20 115 58)(21 59 116)(22 117 60)(23 61 118)(24 119 62)(25 63 120)(26 121 64)(27 49 122)(28 123 50)(29 51 124)(30 125 52)(31 53 126)(32 127 54)(65 83 105)(66 106 84)(67 85 107)(68 108 86)(69 87 109)(70 110 88)(71 89 111)(72 112 90)(73 91 97)(74 98 92)(75 93 99)(76 100 94)(77 95 101)(78 102 96)(79 81 103)(80 104 82)
(1 45 131)(2 46 132)(3 47 133)(4 48 134)(5 33 135)(6 34 136)(7 35 137)(8 36 138)(9 37 139)(10 38 140)(11 39 141)(12 40 142)(13 41 143)(14 42 144)(15 43 129)(16 44 130)(17 55 128)(18 56 113)(19 57 114)(20 58 115)(21 59 116)(22 60 117)(23 61 118)(24 62 119)(25 63 120)(26 64 121)(27 49 122)(28 50 123)(29 51 124)(30 52 125)(31 53 126)(32 54 127)(65 105 83)(66 106 84)(67 107 85)(68 108 86)(69 109 87)(70 110 88)(71 111 89)(72 112 90)(73 97 91)(74 98 92)(75 99 93)(76 100 94)(77 101 95)(78 102 96)(79 103 81)(80 104 82)
(1 119 96)(2 81 120)(3 121 82)(4 83 122)(5 123 84)(6 85 124)(7 125 86)(8 87 126)(9 127 88)(10 89 128)(11 113 90)(12 91 114)(13 115 92)(14 93 116)(15 117 94)(16 95 118)(17 38 71)(18 72 39)(19 40 73)(20 74 41)(21 42 75)(22 76 43)(23 44 77)(24 78 45)(25 46 79)(26 80 47)(27 48 65)(28 66 33)(29 34 67)(30 68 35)(31 36 69)(32 70 37)(49 134 105)(50 106 135)(51 136 107)(52 108 137)(53 138 109)(54 110 139)(55 140 111)(56 112 141)(57 142 97)(58 98 143)(59 144 99)(60 100 129)(61 130 101)(62 102 131)(63 132 103)(64 104 133)
(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)| (17,55,128)(18,113,56)(19,57,114)(20,115,58)(21,59,116)(22,117,60)(23,61,118)(24,119,62)(25,63,120)(26,121,64)(27,49,122)(28,123,50)(29,51,124)(30,125,52)(31,53,126)(32,127,54)(65,83,105)(66,106,84)(67,85,107)(68,108,86)(69,87,109)(70,110,88)(71,89,111)(72,112,90)(73,91,97)(74,98,92)(75,93,99)(76,100,94)(77,95,101)(78,102,96)(79,81,103)(80,104,82), (1,45,131)(2,46,132)(3,47,133)(4,48,134)(5,33,135)(6,34,136)(7,35,137)(8,36,138)(9,37,139)(10,38,140)(11,39,141)(12,40,142)(13,41,143)(14,42,144)(15,43,129)(16,44,130)(17,55,128)(18,56,113)(19,57,114)(20,58,115)(21,59,116)(22,60,117)(23,61,118)(24,62,119)(25,63,120)(26,64,121)(27,49,122)(28,50,123)(29,51,124)(30,52,125)(31,53,126)(32,54,127)(65,105,83)(66,106,84)(67,107,85)(68,108,86)(69,109,87)(70,110,88)(71,111,89)(72,112,90)(73,97,91)(74,98,92)(75,99,93)(76,100,94)(77,101,95)(78,102,96)(79,103,81)(80,104,82), (1,119,96)(2,81,120)(3,121,82)(4,83,122)(5,123,84)(6,85,124)(7,125,86)(8,87,126)(9,127,88)(10,89,128)(11,113,90)(12,91,114)(13,115,92)(14,93,116)(15,117,94)(16,95,118)(17,38,71)(18,72,39)(19,40,73)(20,74,41)(21,42,75)(22,76,43)(23,44,77)(24,78,45)(25,46,79)(26,80,47)(27,48,65)(28,66,33)(29,34,67)(30,68,35)(31,36,69)(32,70,37)(49,134,105)(50,106,135)(51,136,107)(52,108,137)(53,138,109)(54,110,139)(55,140,111)(56,112,141)(57,142,97)(58,98,143)(59,144,99)(60,100,129)(61,130,101)(62,102,131)(63,132,103)(64,104,133), (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( (17,55,128)(18,113,56)(19,57,114)(20,115,58)(21,59,116)(22,117,60)(23,61,118)(24,119,62)(25,63,120)(26,121,64)(27,49,122)(28,123,50)(29,51,124)(30,125,52)(31,53,126)(32,127,54)(65,83,105)(66,106,84)(67,85,107)(68,108,86)(69,87,109)(70,110,88)(71,89,111)(72,112,90)(73,91,97)(74,98,92)(75,93,99)(76,100,94)(77,95,101)(78,102,96)(79,81,103)(80,104,82), (1,45,131)(2,46,132)(3,47,133)(4,48,134)(5,33,135)(6,34,136)(7,35,137)(8,36,138)(9,37,139)(10,38,140)(11,39,141)(12,40,142)(13,41,143)(14,42,144)(15,43,129)(16,44,130)(17,55,128)(18,56,113)(19,57,114)(20,58,115)(21,59,116)(22,60,117)(23,61,118)(24,62,119)(25,63,120)(26,64,121)(27,49,122)(28,50,123)(29,51,124)(30,52,125)(31,53,126)(32,54,127)(65,105,83)(66,106,84)(67,107,85)(68,108,86)(69,109,87)(70,110,88)(71,111,89)(72,112,90)(73,97,91)(74,98,92)(75,99,93)(76,100,94)(77,101,95)(78,102,96)(79,103,81)(80,104,82), (1,119,96)(2,81,120)(3,121,82)(4,83,122)(5,123,84)(6,85,124)(7,125,86)(8,87,126)(9,127,88)(10,89,128)(11,113,90)(12,91,114)(13,115,92)(14,93,116)(15,117,94)(16,95,118)(17,38,71)(18,72,39)(19,40,73)(20,74,41)(21,42,75)(22,76,43)(23,44,77)(24,78,45)(25,46,79)(26,80,47)(27,48,65)(28,66,33)(29,34,67)(30,68,35)(31,36,69)(32,70,37)(49,134,105)(50,106,135)(51,136,107)(52,108,137)(53,138,109)(54,110,139)(55,140,111)(56,112,141)(57,142,97)(58,98,143)(59,144,99)(60,100,129)(61,130,101)(62,102,131)(63,132,103)(64,104,133), (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([[(17,55,128),(18,113,56),(19,57,114),(20,115,58),(21,59,116),(22,117,60),(23,61,118),(24,119,62),(25,63,120),(26,121,64),(27,49,122),(28,123,50),(29,51,124),(30,125,52),(31,53,126),(32,127,54),(65,83,105),(66,106,84),(67,85,107),(68,108,86),(69,87,109),(70,110,88),(71,89,111),(72,112,90),(73,91,97),(74,98,92),(75,93,99),(76,100,94),(77,95,101),(78,102,96),(79,81,103),(80,104,82)], [(1,45,131),(2,46,132),(3,47,133),(4,48,134),(5,33,135),(6,34,136),(7,35,137),(8,36,138),(9,37,139),(10,38,140),(11,39,141),(12,40,142),(13,41,143),(14,42,144),(15,43,129),(16,44,130),(17,55,128),(18,56,113),(19,57,114),(20,58,115),(21,59,116),(22,60,117),(23,61,118),(24,62,119),(25,63,120),(26,64,121),(27,49,122),(28,50,123),(29,51,124),(30,52,125),(31,53,126),(32,54,127),(65,105,83),(66,106,84),(67,107,85),(68,108,86),(69,109,87),(70,110,88),(71,111,89),(72,112,90),(73,97,91),(74,98,92),(75,99,93),(76,100,94),(77,101,95),(78,102,96),(79,103,81),(80,104,82)], [(1,119,96),(2,81,120),(3,121,82),(4,83,122),(5,123,84),(6,85,124),(7,125,86),(8,87,126),(9,127,88),(10,89,128),(11,113,90),(12,91,114),(13,115,92),(14,93,116),(15,117,94),(16,95,118),(17,38,71),(18,72,39),(19,40,73),(20,74,41),(21,42,75),(22,76,43),(23,44,77),(24,78,45),(25,46,79),(26,80,47),(27,48,65),(28,66,33),(29,34,67),(30,68,35),(31,36,69),(32,70,37),(49,134,105),(50,106,135),(51,136,107),(52,108,137),(53,138,109),(54,110,139),(55,140,111),(56,112,141),(57,142,97),(58,98,143),(59,144,99),(60,100,129),(61,130,101),(62,102,131),(63,132,103),(64,104,133)], [(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)]])
80 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 16A | ··· | 16H | 24A | ··· | 24H | 24I | ··· | 24X | 48A | ··· | 48P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 9 | ··· | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | S3 | Dic3 | C3⋊C8 | C3⋊C16 | He3⋊C2 | He3⋊3C4 | He3⋊4C8 | He3⋊4C16 |
| kernel | He3⋊4C16 | C8×He3 | C4×He3 | C2×He3 | He3 | C3×C24 | C3×C12 | C3×C6 | C32 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 8 | 16 | 4 | 4 | 8 | 16 |
Matrix representation of He3⋊4C16 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 61 |
| 35 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 35 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 12 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(97))| [1,0,0,0,35,0,0,0,61],[35,0,0,0,35,0,0,0,35],[0,0,1,1,0,0,0,1,0],[12,0,0,0,0,12,0,12,0] >;
He3⋊4C16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_4C_{16} % in TeX
G:=Group("He3:4C16"); // GroupNames label
G:=SmallGroup(432,33);
// by ID
G=gap.SmallGroup(432,33);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,14,36,58,1124,4037,537]);
// 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^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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