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## G = C16×He3order 432 = 24·33

### Direct product of C16 and He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C16×He3
 Chief series C1 — C2 — C4 — C12 — C24 — C3×C24 — C8×He3 — C16×He3
 Lower central C1 — C3 — C16×He3
 Upper central C1 — C48 — C16×He3

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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