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G = He3⋊2C16order 432 = 24·33

The semidirect product of He3 and C16 acting via C16/C4=C4

Aliases: He32C16, C4.2(He3⋊C4), C2.(He32C8), (C2×He3).2C8, (C4×He3).1C4, C12.7(C32⋊C4), C3.(C322C16), He34C8.1C2, C6.2(C322C8), SmallGroup(432,57)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊2C16
 Chief series C1 — C3 — He3 — C2×He3 — C4×He3 — He3⋊4C8 — He3⋊2C16
 Lower central He3 — He3⋊2C16
 Upper central C1 — C12

Generators and relations for He32C16
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 He32C16
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 He32C16 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] >;

He32C16 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

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