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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 [×2], C3 [×4], C4, C4, C22, S3 [×8], C6 [×4], C8, C8, C2×C4, C32, Dic3 [×4], C12 [×4], D6 [×4], C16, C16, C2×C8, C3⋊S3 [×2], C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×4], C2×C16, C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16 [×4], C48 [×4], S3×C8 [×4], C324C8, C3×C24, C4×C3⋊S3, S3×C16 [×4], C24.S3, C3×C48, C8×C3⋊S3, C16×C3⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, D6 [×4], C16 [×2], C2×C8, C3⋊S3, C4×S3 [×4], C2×C16, C2×C3⋊S3, S3×C8 [×4], C4×C3⋊S3, S3×C16 [×4], 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 42 110)(2 43 111)(3 44 112)(4 45 97)(5 46 98)(6 47 99)(7 48 100)(8 33 101)(9 34 102)(10 35 103)(11 36 104)(12 37 105)(13 38 106)(14 39 107)(15 40 108)(16 41 109)(17 144 56)(18 129 57)(19 130 58)(20 131 59)(21 132 60)(22 133 61)(23 134 62)(24 135 63)(25 136 64)(26 137 49)(27 138 50)(28 139 51)(29 140 52)(30 141 53)(31 142 54)(32 143 55)(65 122 89)(66 123 90)(67 124 91)(68 125 92)(69 126 93)(70 127 94)(71 128 95)(72 113 96)(73 114 81)(74 115 82)(75 116 83)(76 117 84)(77 118 85)(78 119 86)(79 120 87)(80 121 88)
(1 116 135)(2 117 136)(3 118 137)(4 119 138)(5 120 139)(6 121 140)(7 122 141)(8 123 142)(9 124 143)(10 125 144)(11 126 129)(12 127 130)(13 128 131)(14 113 132)(15 114 133)(16 115 134)(17 103 68)(18 104 69)(19 105 70)(20 106 71)(21 107 72)(22 108 73)(23 109 74)(24 110 75)(25 111 76)(26 112 77)(27 97 78)(28 98 79)(29 99 80)(30 100 65)(31 101 66)(32 102 67)(33 90 54)(34 91 55)(35 92 56)(36 93 57)(37 94 58)(38 95 59)(39 96 60)(40 81 61)(41 82 62)(42 83 63)(43 84 64)(44 85 49)(45 86 50)(46 87 51)(47 88 52)(48 89 53)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 81)(31 82)(32 83)(33 109)(34 110)(35 111)(36 112)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 65)(62 66)(63 67)(64 68)(113 140)(114 141)(115 142)(116 143)(117 144)(118 129)(119 130)(120 131)(121 132)(122 133)(123 134)(124 135)(125 136)(126 137)(127 138)(128 139)

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

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

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

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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