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## G = C16×C3⋊S3order 288 = 25·32

### Direct product of C16 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C16×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C8×C3⋊S3 — C16×C3⋊S3
 Lower central C32 — C16×C3⋊S3
 Upper central C1 — C16

Generators and relations for C16×C3⋊S3
G = < a,b,c,d | a16=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 228 in 84 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, C16, C16, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C2×C16, C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16, C48, S3×C8, C324C8, C3×C24, C4×C3⋊S3, S3×C16, C24.S3, C3×C48, C8×C3⋊S3, C16×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C16, C2×C8, C3⋊S3, C4×S3, C2×C16, C2×C3⋊S3, S3×C8, C4×C3⋊S3, S3×C16, C8×C3⋊S3, C16×C3⋊S3

Smallest permutation representation of C16×C3⋊S3
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 33 137)(2 34 138)(3 35 139)(4 36 140)(5 37 141)(6 38 142)(7 39 143)(8 40 144)(9 41 129)(10 42 130)(11 43 131)(12 44 132)(13 45 133)(14 46 134)(15 47 135)(16 48 136)(17 69 105)(18 70 106)(19 71 107)(20 72 108)(21 73 109)(22 74 110)(23 75 111)(24 76 112)(25 77 97)(26 78 98)(27 79 99)(28 80 100)(29 65 101)(30 66 102)(31 67 103)(32 68 104)(49 88 126)(50 89 127)(51 90 128)(52 91 113)(53 92 114)(54 93 115)(55 94 116)(56 95 117)(57 96 118)(58 81 119)(59 82 120)(60 83 121)(61 84 122)(62 85 123)(63 86 124)(64 87 125)
(1 90 68)(2 91 69)(3 92 70)(4 93 71)(5 94 72)(6 95 73)(7 96 74)(8 81 75)(9 82 76)(10 83 77)(11 84 78)(12 85 79)(13 86 80)(14 87 65)(15 88 66)(16 89 67)(17 138 52)(18 139 53)(19 140 54)(20 141 55)(21 142 56)(22 143 57)(23 144 58)(24 129 59)(25 130 60)(26 131 61)(27 132 62)(28 133 63)(29 134 64)(30 135 49)(31 136 50)(32 137 51)(33 128 104)(34 113 105)(35 114 106)(36 115 107)(37 116 108)(38 117 109)(39 118 110)(40 119 111)(41 120 112)(42 121 97)(43 122 98)(44 123 99)(45 124 100)(46 125 101)(47 126 102)(48 127 103)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 110)(50 111)(51 112)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(61 106)(62 107)(63 108)(64 109)(65 95)(66 96)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 16A ··· 16H 16I ··· 16P 24A ··· 24P 48A ··· 48AF order 1 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 16 ··· 16 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 9 9 2 2 2 2 1 1 9 9 2 2 2 2 1 1 1 1 9 9 9 9 2 ··· 2 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 C16 S3 D6 C4×S3 S3×C8 S3×C16 kernel C16×C3⋊S3 C24.S3 C3×C48 C8×C3⋊S3 C32⋊4C8 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3⋊S3 C48 C24 C12 C6 C3 # reps 1 1 1 1 2 2 4 4 16 4 4 8 16 32

Matrix representation of C16×C3⋊S3 in GL5(𝔽97)

 8 0 0 0 0 0 22 0 0 0 0 0 22 0 0 0 0 0 33 0 0 0 0 0 33
,
 1 0 0 0 0 0 96 1 0 0 0 96 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 95 94 0 0 0 1 1
,
 1 0 0 0 0 0 1 96 0 0 0 0 96 0 0 0 0 0 1 0 0 0 0 96 96

G:=sub<GL(5,GF(97))| [8,0,0,0,0,0,22,0,0,0,0,0,22,0,0,0,0,0,33,0,0,0,0,0,33],[1,0,0,0,0,0,96,96,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,95,1,0,0,0,94,1],[1,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,1,96,0,0,0,0,96] >;

C16×C3⋊S3 in GAP, Magma, Sage, TeX

C_{16}\times C_3\rtimes S_3
% in TeX

G:=Group("C16xC3:S3");
// GroupNames label

G:=SmallGroup(288,272);
// by ID

G=gap.SmallGroup(288,272);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,36,58,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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