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G = C325D16order 288 = 25·32

2nd semidirect product of C32 and D16 acting via D16/C16=C2

metabelian, supersoluble, monomial

Aliases: C481S3, C31D48, C6.7D24, C325D16, C24.72D6, C12.43D12, (C3×C48)⋊1C2, C161(C3⋊S3), (C3×C6).23D8, C325D81C2, (C3×C12).118D4, C4.1(C12⋊S3), C2.3(C325D8), (C3×C24).50C22, C8.13(C2×C3⋊S3), SmallGroup(288,274)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C325D16
C1C3C32C3×C6C3×C12C3×C24C325D8 — C325D16
C32C3×C6C3×C12C3×C24 — C325D16
C1C2C4C8C16

Generators and relations for C325D16
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 712 in 84 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C22 [×2], S3 [×8], C6 [×4], C8, D4 [×2], C32, C12 [×4], D6 [×8], C16, D8 [×2], C3⋊S3 [×2], C3×C6, C24 [×4], D12 [×8], D16, C3×C12, C2×C3⋊S3 [×2], C48 [×4], D24 [×8], C3×C24, C12⋊S3 [×2], D48 [×4], C3×C48, C325D8 [×2], C325D16
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, D12 [×4], D16, C2×C3⋊S3, D24 [×4], C12⋊S3, D48 [×4], C325D8, C325D16

Smallest permutation representation of C325D16
On 144 points
Generators in S144
(1 38 134)(2 39 135)(3 40 136)(4 41 137)(5 42 138)(6 43 139)(7 44 140)(8 45 141)(9 46 142)(10 47 143)(11 48 144)(12 33 129)(13 34 130)(14 35 131)(15 36 132)(16 37 133)(17 127 108)(18 128 109)(19 113 110)(20 114 111)(21 115 112)(22 116 97)(23 117 98)(24 118 99)(25 119 100)(26 120 101)(27 121 102)(28 122 103)(29 123 104)(30 124 105)(31 125 106)(32 126 107)(49 69 95)(50 70 96)(51 71 81)(52 72 82)(53 73 83)(54 74 84)(55 75 85)(56 76 86)(57 77 87)(58 78 88)(59 79 89)(60 80 90)(61 65 91)(62 66 92)(63 67 93)(64 68 94)
(1 90 106)(2 91 107)(3 92 108)(4 93 109)(5 94 110)(6 95 111)(7 96 112)(8 81 97)(9 82 98)(10 83 99)(11 84 100)(12 85 101)(13 86 102)(14 87 103)(15 88 104)(16 89 105)(17 40 62)(18 41 63)(19 42 64)(20 43 49)(21 44 50)(22 45 51)(23 46 52)(24 47 53)(25 48 54)(26 33 55)(27 34 56)(28 35 57)(29 36 58)(30 37 59)(31 38 60)(32 39 61)(65 126 135)(66 127 136)(67 128 137)(68 113 138)(69 114 139)(70 115 140)(71 116 141)(72 117 142)(73 118 143)(74 119 144)(75 120 129)(76 121 130)(77 122 131)(78 123 132)(79 124 133)(80 125 134)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 77)(18 76)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 69)(26 68)(27 67)(28 66)(29 65)(30 80)(31 79)(32 78)(33 138)(34 137)(35 136)(36 135)(37 134)(38 133)(39 132)(40 131)(41 130)(42 129)(43 144)(44 143)(45 142)(46 141)(47 140)(48 139)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 128)(57 127)(58 126)(59 125)(60 124)(61 123)(62 122)(63 121)(64 120)(81 98)(82 97)(83 112)(84 111)(85 110)(86 109)(87 108)(88 107)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 99)

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

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

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

75 conjugacy classes

class 1 2A2B2C3A3B3C3D 4 6A6B6C6D8A8B12A···12H16A16B16C16D24A···24P48A···48AF
order12223333466668812···121616161624···2448···48
size117272222222222222···222222···22···2

75 irreducible representations

dim11122222222
type+++++++++++
imageC1C2C2S3D4D6D8D12D16D24D48
kernelC325D16C3×C48C325D8C48C3×C12C24C3×C6C12C32C6C3
# reps1124142841632

Matrix representation of C325D16 in GL4(𝔽97) generated by

1000
0100
0001
009696
,
0100
969600
0001
009696
,
657800
198400
006839
005829
,
657800
133200
002968
003968
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,0,96,0,0,1,96],[0,96,0,0,1,96,0,0,0,0,0,96,0,0,1,96],[65,19,0,0,78,84,0,0,0,0,68,58,0,0,39,29],[65,13,0,0,78,32,0,0,0,0,29,39,0,0,68,68] >;

C325D16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_5D_{16}
% in TeX

G:=Group("C3^2:5D16");
// GroupNames label

G:=SmallGroup(288,274);
// by ID

G=gap.SmallGroup(288,274);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,254,142,675,80,2693,9414]);
// Polycyclic

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

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