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## G = He3⋊C16order 432 = 24·33

### The semidirect product of He3 and C16 acting via C16/C2=C8

Aliases: He3⋊C16, C6.1F9, C2.(He3⋊C8), (C2×He3).C8, C3.(C2.F9), He33C4.C4, He32C8.2C2, SmallGroup(432,233)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C16
 Chief series C1 — C3 — He3 — C2×He3 — He3⋊3C4 — He3⋊2C8 — He3⋊C16
 Lower central He3 — He3⋊C16
 Upper central C1 — C2

Generators and relations for He3⋊C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, cac-1=ab-1, dad-1=c, bc=cb, dbd-1=b-1, dcd-1=ab-1c >

Character table of He3⋊C16

 class 1 2 3A 3B 4A 4B 6A 6B 8A 8B 8C 8D 12A 12B 16A 16B 16C 16D 16E 16F 16G 16H 24A 24B 24C 24D size 1 1 2 24 9 9 2 24 9 9 9 9 18 18 27 27 27 27 27 27 27 27 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -i i i i -i -i -i i -1 -1 -1 -1 linear of order 4 ρ4 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 i -i -i -i i i i -i -1 -1 -1 -1 linear of order 4 ρ5 1 1 1 1 -1 -1 1 1 i -i i -i -1 -1 ζ85 ζ83 ζ87 ζ87 ζ8 ζ8 ζ85 ζ83 -i i -i i linear of order 8 ρ6 1 1 1 1 -1 -1 1 1 -i i -i i -1 -1 ζ83 ζ85 ζ8 ζ8 ζ87 ζ87 ζ83 ζ85 i -i i -i linear of order 8 ρ7 1 1 1 1 -1 -1 1 1 i -i i -i -1 -1 ζ8 ζ87 ζ83 ζ83 ζ85 ζ85 ζ8 ζ87 -i i -i i linear of order 8 ρ8 1 1 1 1 -1 -1 1 1 -i i -i i -1 -1 ζ87 ζ8 ζ85 ζ85 ζ83 ζ83 ζ87 ζ8 i -i i -i linear of order 8 ρ9 1 -1 1 1 -i i -1 -1 ζ1614 ζ1610 ζ166 ζ162 i -i ζ167 ζ16 ζ1613 ζ165 ζ1611 ζ163 ζ1615 ζ169 ζ162 ζ166 ζ1610 ζ1614 linear of order 16 ρ10 1 -1 1 1 -i i -1 -1 ζ166 ζ162 ζ1614 ζ1610 i -i ζ163 ζ165 ζ16 ζ169 ζ167 ζ1615 ζ1611 ζ1613 ζ1610 ζ1614 ζ162 ζ166 linear of order 16 ρ11 1 -1 1 1 i -i -1 -1 ζ162 ζ166 ζ1610 ζ1614 -i i ζ169 ζ1615 ζ163 ζ1611 ζ165 ζ1613 ζ16 ζ167 ζ1614 ζ1610 ζ166 ζ162 linear of order 16 ρ12 1 -1 1 1 i -i -1 -1 ζ1610 ζ1614 ζ162 ζ166 -i i ζ1613 ζ1611 ζ1615 ζ167 ζ169 ζ16 ζ165 ζ163 ζ166 ζ162 ζ1614 ζ1610 linear of order 16 ρ13 1 -1 1 1 -i i -1 -1 ζ166 ζ162 ζ1614 ζ1610 i -i ζ1611 ζ1613 ζ169 ζ16 ζ1615 ζ167 ζ163 ζ165 ζ1610 ζ1614 ζ162 ζ166 linear of order 16 ρ14 1 -1 1 1 i -i -1 -1 ζ162 ζ166 ζ1610 ζ1614 -i i ζ16 ζ167 ζ1611 ζ163 ζ1613 ζ165 ζ169 ζ1615 ζ1614 ζ1610 ζ166 ζ162 linear of order 16 ρ15 1 -1 1 1 i -i -1 -1 ζ1610 ζ1614 ζ162 ζ166 -i i ζ165 ζ163 ζ167 ζ1615 ζ16 ζ169 ζ1613 ζ1611 ζ166 ζ162 ζ1614 ζ1610 linear of order 16 ρ16 1 -1 1 1 -i i -1 -1 ζ1614 ζ1610 ζ166 ζ162 i -i ζ1615 ζ169 ζ165 ζ1613 ζ163 ζ1611 ζ167 ζ16 ζ162 ζ166 ζ1610 ζ1614 linear of order 16 ρ17 6 6 -3 0 -2 -2 -3 0 2 2 2 2 1 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from He3⋊C8 ρ18 6 6 -3 0 -2 -2 -3 0 -2 -2 -2 -2 1 1 0 0 0 0 0 0 0 0 1 1 1 1 symplectic lifted from He3⋊C8, Schur index 2 ρ19 6 6 -3 0 2 2 -3 0 2i -2i 2i -2i -1 -1 0 0 0 0 0 0 0 0 i -i i -i complex lifted from He3⋊C8 ρ20 6 6 -3 0 2 2 -3 0 -2i 2i -2i 2i -1 -1 0 0 0 0 0 0 0 0 -i i -i i complex lifted from He3⋊C8 ρ21 6 -6 -3 0 -2i 2i 3 0 2ζ8 2ζ83 2ζ85 2ζ87 -i i 0 0 0 0 0 0 0 0 ζ83 ζ8 ζ87 ζ85 complex faithful, Schur index 2 ρ22 6 -6 -3 0 2i -2i 3 0 2ζ87 2ζ85 2ζ83 2ζ8 i -i 0 0 0 0 0 0 0 0 ζ85 ζ87 ζ8 ζ83 complex faithful, Schur index 2 ρ23 6 -6 -3 0 2i -2i 3 0 2ζ83 2ζ8 2ζ87 2ζ85 i -i 0 0 0 0 0 0 0 0 ζ8 ζ83 ζ85 ζ87 complex faithful, Schur index 2 ρ24 6 -6 -3 0 -2i 2i 3 0 2ζ85 2ζ87 2ζ8 2ζ83 -i i 0 0 0 0 0 0 0 0 ζ87 ζ85 ζ83 ζ8 complex faithful, Schur index 2 ρ25 8 8 8 -1 0 0 8 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F9 ρ26 8 -8 8 -1 0 0 -8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C2.F9, Schur index 2

Smallest permutation representation of He3⋊C16
On 144 points
Generators in S144
(1 35 117)(3 37 112)(4 58 97)(5 84 59)(6 20 133)(7 123 61)(8 87 42)(9 43 125)(11 45 104)(12 50 105)(13 92 51)(14 28 141)(15 115 53)(16 95 34)(17 57 82)(18 79 83)(19 121 80)(21 100 66)(22 124 62)(23 63 102)(25 49 90)(26 71 91)(27 113 72)(29 108 74)(30 116 54)(31 55 110)(33 142 94)(36 56 77)(38 120 131)(39 132 98)(41 134 86)(44 64 69)(46 128 139)(47 140 106)(67 135 101)(68 88 136)(70 127 138)(75 143 109)(76 96 144)(78 119 130)(81 111 118)(85 99 122)(89 103 126)(93 107 114)
(1 31 144)(2 129 32)(3 17 130)(4 131 18)(5 19 132)(6 133 20)(7 21 134)(8 135 22)(9 23 136)(10 137 24)(11 25 138)(12 139 26)(13 27 140)(14 141 28)(15 29 142)(16 143 30)(33 53 74)(34 75 54)(35 55 76)(36 77 56)(37 57 78)(38 79 58)(39 59 80)(40 65 60)(41 61 66)(42 67 62)(43 63 68)(44 69 64)(45 49 70)(46 71 50)(47 51 72)(48 73 52)(81 111 118)(82 119 112)(83 97 120)(84 121 98)(85 99 122)(86 123 100)(87 101 124)(88 125 102)(89 103 126)(90 127 104)(91 105 128)(92 113 106)(93 107 114)(94 115 108)(95 109 116)(96 117 110)
(2 36 111)(3 57 112)(4 83 58)(5 19 132)(6 122 60)(7 86 41)(8 42 124)(10 44 103)(11 49 104)(12 91 50)(13 27 140)(14 114 52)(15 94 33)(16 34 116)(17 78 82)(18 120 79)(20 99 65)(21 123 61)(22 62 101)(24 64 89)(25 70 90)(26 128 71)(28 107 73)(29 115 53)(30 54 109)(32 56 81)(35 55 76)(37 119 130)(38 131 97)(40 133 85)(43 63 68)(45 127 138)(46 139 105)(48 141 93)(66 134 100)(67 87 135)(69 126 137)(74 142 108)(75 95 143)(77 118 129)(84 98 121)(88 102 125)(92 106 113)(96 110 117)
(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)| (1,35,117)(3,37,112)(4,58,97)(5,84,59)(6,20,133)(7,123,61)(8,87,42)(9,43,125)(11,45,104)(12,50,105)(13,92,51)(14,28,141)(15,115,53)(16,95,34)(17,57,82)(18,79,83)(19,121,80)(21,100,66)(22,124,62)(23,63,102)(25,49,90)(26,71,91)(27,113,72)(29,108,74)(30,116,54)(31,55,110)(33,142,94)(36,56,77)(38,120,131)(39,132,98)(41,134,86)(44,64,69)(46,128,139)(47,140,106)(67,135,101)(68,88,136)(70,127,138)(75,143,109)(76,96,144)(78,119,130)(81,111,118)(85,99,122)(89,103,126)(93,107,114), (1,31,144)(2,129,32)(3,17,130)(4,131,18)(5,19,132)(6,133,20)(7,21,134)(8,135,22)(9,23,136)(10,137,24)(11,25,138)(12,139,26)(13,27,140)(14,141,28)(15,29,142)(16,143,30)(33,53,74)(34,75,54)(35,55,76)(36,77,56)(37,57,78)(38,79,58)(39,59,80)(40,65,60)(41,61,66)(42,67,62)(43,63,68)(44,69,64)(45,49,70)(46,71,50)(47,51,72)(48,73,52)(81,111,118)(82,119,112)(83,97,120)(84,121,98)(85,99,122)(86,123,100)(87,101,124)(88,125,102)(89,103,126)(90,127,104)(91,105,128)(92,113,106)(93,107,114)(94,115,108)(95,109,116)(96,117,110), (2,36,111)(3,57,112)(4,83,58)(5,19,132)(6,122,60)(7,86,41)(8,42,124)(10,44,103)(11,49,104)(12,91,50)(13,27,140)(14,114,52)(15,94,33)(16,34,116)(17,78,82)(18,120,79)(20,99,65)(21,123,61)(22,62,101)(24,64,89)(25,70,90)(26,128,71)(28,107,73)(29,115,53)(30,54,109)(32,56,81)(35,55,76)(37,119,130)(38,131,97)(40,133,85)(43,63,68)(45,127,138)(46,139,105)(48,141,93)(66,134,100)(67,87,135)(69,126,137)(74,142,108)(75,95,143)(77,118,129)(84,98,121)(88,102,125)(92,106,113)(96,110,117), (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( (1,35,117)(3,37,112)(4,58,97)(5,84,59)(6,20,133)(7,123,61)(8,87,42)(9,43,125)(11,45,104)(12,50,105)(13,92,51)(14,28,141)(15,115,53)(16,95,34)(17,57,82)(18,79,83)(19,121,80)(21,100,66)(22,124,62)(23,63,102)(25,49,90)(26,71,91)(27,113,72)(29,108,74)(30,116,54)(31,55,110)(33,142,94)(36,56,77)(38,120,131)(39,132,98)(41,134,86)(44,64,69)(46,128,139)(47,140,106)(67,135,101)(68,88,136)(70,127,138)(75,143,109)(76,96,144)(78,119,130)(81,111,118)(85,99,122)(89,103,126)(93,107,114), (1,31,144)(2,129,32)(3,17,130)(4,131,18)(5,19,132)(6,133,20)(7,21,134)(8,135,22)(9,23,136)(10,137,24)(11,25,138)(12,139,26)(13,27,140)(14,141,28)(15,29,142)(16,143,30)(33,53,74)(34,75,54)(35,55,76)(36,77,56)(37,57,78)(38,79,58)(39,59,80)(40,65,60)(41,61,66)(42,67,62)(43,63,68)(44,69,64)(45,49,70)(46,71,50)(47,51,72)(48,73,52)(81,111,118)(82,119,112)(83,97,120)(84,121,98)(85,99,122)(86,123,100)(87,101,124)(88,125,102)(89,103,126)(90,127,104)(91,105,128)(92,113,106)(93,107,114)(94,115,108)(95,109,116)(96,117,110), (2,36,111)(3,57,112)(4,83,58)(5,19,132)(6,122,60)(7,86,41)(8,42,124)(10,44,103)(11,49,104)(12,91,50)(13,27,140)(14,114,52)(15,94,33)(16,34,116)(17,78,82)(18,120,79)(20,99,65)(21,123,61)(22,62,101)(24,64,89)(25,70,90)(26,128,71)(28,107,73)(29,115,53)(30,54,109)(32,56,81)(35,55,76)(37,119,130)(38,131,97)(40,133,85)(43,63,68)(45,127,138)(46,139,105)(48,141,93)(66,134,100)(67,87,135)(69,126,137)(74,142,108)(75,95,143)(77,118,129)(84,98,121)(88,102,125)(92,106,113)(96,110,117), (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([[(1,35,117),(3,37,112),(4,58,97),(5,84,59),(6,20,133),(7,123,61),(8,87,42),(9,43,125),(11,45,104),(12,50,105),(13,92,51),(14,28,141),(15,115,53),(16,95,34),(17,57,82),(18,79,83),(19,121,80),(21,100,66),(22,124,62),(23,63,102),(25,49,90),(26,71,91),(27,113,72),(29,108,74),(30,116,54),(31,55,110),(33,142,94),(36,56,77),(38,120,131),(39,132,98),(41,134,86),(44,64,69),(46,128,139),(47,140,106),(67,135,101),(68,88,136),(70,127,138),(75,143,109),(76,96,144),(78,119,130),(81,111,118),(85,99,122),(89,103,126),(93,107,114)], [(1,31,144),(2,129,32),(3,17,130),(4,131,18),(5,19,132),(6,133,20),(7,21,134),(8,135,22),(9,23,136),(10,137,24),(11,25,138),(12,139,26),(13,27,140),(14,141,28),(15,29,142),(16,143,30),(33,53,74),(34,75,54),(35,55,76),(36,77,56),(37,57,78),(38,79,58),(39,59,80),(40,65,60),(41,61,66),(42,67,62),(43,63,68),(44,69,64),(45,49,70),(46,71,50),(47,51,72),(48,73,52),(81,111,118),(82,119,112),(83,97,120),(84,121,98),(85,99,122),(86,123,100),(87,101,124),(88,125,102),(89,103,126),(90,127,104),(91,105,128),(92,113,106),(93,107,114),(94,115,108),(95,109,116),(96,117,110)], [(2,36,111),(3,57,112),(4,83,58),(5,19,132),(6,122,60),(7,86,41),(8,42,124),(10,44,103),(11,49,104),(12,91,50),(13,27,140),(14,114,52),(15,94,33),(16,34,116),(17,78,82),(18,120,79),(20,99,65),(21,123,61),(22,62,101),(24,64,89),(25,70,90),(26,128,71),(28,107,73),(29,115,53),(30,54,109),(32,56,81),(35,55,76),(37,119,130),(38,131,97),(40,133,85),(43,63,68),(45,127,138),(46,139,105),(48,141,93),(66,134,100),(67,87,135),(69,126,137),(74,142,108),(75,95,143),(77,118,129),(84,98,121),(88,102,125),(92,106,113),(96,110,117)], [(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)]])

Matrix representation of He3⋊C16 in GL6(𝔽97)

 1 1 96 95 0 0 0 0 1 96 0 0 0 0 0 96 0 1 1 1 0 96 96 96 1 0 0 96 0 0 0 1 0 96 0 0
,
 0 96 0 0 0 0 1 96 0 0 0 0 1 0 96 96 0 0 0 96 1 0 0 0 1 0 0 0 96 96 0 96 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 96 96 0 0 1 1 0 0 96 96 0 0 0 0 1 0
,
 58 22 75 39 75 39 36 58 39 61 39 61 0 22 75 39 0 0 58 58 39 61 75 39 36 0 39 61 0 0 0 22 0 0 75 39

G:=sub<GL(6,GF(97))| [1,0,0,1,1,0,1,0,0,1,0,1,96,1,0,0,0,0,95,96,96,96,96,96,0,0,0,96,0,0,0,0,1,96,0,0],[0,1,1,0,1,0,96,96,0,96,0,96,0,0,96,1,0,0,0,0,96,0,0,0,0,0,0,0,96,1,0,0,0,0,96,0],[1,0,0,1,1,0,0,1,0,1,1,0,0,0,0,96,0,0,0,0,1,96,0,0,0,0,0,0,96,1,0,0,0,0,96,0],[58,36,0,58,36,0,22,58,22,58,0,22,75,39,75,39,39,0,39,61,39,61,61,0,75,39,0,75,0,75,39,61,0,39,0,39] >;

He3⋊C16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_{16}
% in TeX

G:=Group("He3:C16");
// GroupNames label

G:=SmallGroup(432,233);
// by ID

G=gap.SmallGroup(432,233);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,14,36,58,1684,1971,1558,4709,8748,4051,915,14118]);
// 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=c,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a*b^-1*c>;
// generators/relations

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