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## G = He3⋊4C16order 432 = 24·33

### 2nd semidirect product of He3 and C16 acting via C16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊4C16
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — C8×He3 — He3⋊4C16
 Lower central He3 — He3⋊4C16
 Upper central C1 — C24

Generators and relations for He34C16
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 He34C16
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 He34C16 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] >;

He34C16 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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