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G = C327D16order 288 = 25·32

2nd semidirect product of C32 and D16 acting via D16/D8=C2

metabelian, supersoluble, monomial

Aliases: C327D16, C24.18D6, (C3×D8)⋊1S3, D81(C3⋊S3), (C3×C6).37D8, C325D85C2, C33(C3⋊D16), (C3×C12).52D4, (C32×D8)⋊2C2, C24.S33C2, C6.23(D4⋊S3), C12.34(C3⋊D4), (C3×C24).17C22, C4.1(C327D4), C2.4(C327D8), C8.4(C2×C3⋊S3), SmallGroup(288,301)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C327D16
C1C3C32C3×C6C3×C12C3×C24C325D8 — C327D16
C32C3×C6C3×C12C3×C24 — C327D16
C1C2C4C8D8

Generators and relations for C327D16
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 488 in 84 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C22 [×2], S3 [×4], C6 [×4], C6 [×4], C8, D4 [×2], C32, C12 [×4], D6 [×4], C2×C6 [×4], C16, D8, D8, C3⋊S3, C3×C6, C3×C6, C24 [×4], D12 [×4], C3×D4 [×4], D16, C3×C12, C2×C3⋊S3, C62, C3⋊C16 [×4], D24 [×4], C3×D8 [×4], C3×C24, C12⋊S3, D4×C32, C3⋊D16 [×4], C24.S3, C325D8, C32×D8, C327D16
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, C3⋊D4 [×4], D16, C2×C3⋊S3, D4⋊S3 [×4], C327D4, C3⋊D16 [×4], C327D8, C327D16

Smallest permutation representation of C327D16
On 144 points
Generators in S144
(1 99 63)(2 64 100)(3 101 49)(4 50 102)(5 103 51)(6 52 104)(7 105 53)(8 54 106)(9 107 55)(10 56 108)(11 109 57)(12 58 110)(13 111 59)(14 60 112)(15 97 61)(16 62 98)(17 74 46)(18 47 75)(19 76 48)(20 33 77)(21 78 34)(22 35 79)(23 80 36)(24 37 65)(25 66 38)(26 39 67)(27 68 40)(28 41 69)(29 70 42)(30 43 71)(31 72 44)(32 45 73)(81 138 116)(82 117 139)(83 140 118)(84 119 141)(85 142 120)(86 121 143)(87 144 122)(88 123 129)(89 130 124)(90 125 131)(91 132 126)(92 127 133)(93 134 128)(94 113 135)(95 136 114)(96 115 137)
(1 33 122)(2 123 34)(3 35 124)(4 125 36)(5 37 126)(6 127 38)(7 39 128)(8 113 40)(9 41 114)(10 115 42)(11 43 116)(12 117 44)(13 45 118)(14 119 46)(15 47 120)(16 121 48)(17 60 141)(18 142 61)(19 62 143)(20 144 63)(21 64 129)(22 130 49)(23 50 131)(24 132 51)(25 52 133)(26 134 53)(27 54 135)(28 136 55)(29 56 137)(30 138 57)(31 58 139)(32 140 59)(65 91 103)(66 104 92)(67 93 105)(68 106 94)(69 95 107)(70 108 96)(71 81 109)(72 110 82)(73 83 111)(74 112 84)(75 85 97)(76 98 86)(77 87 99)(78 100 88)(79 89 101)(80 102 90)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 122)(34 121)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 128)(44 127)(45 126)(46 125)(47 124)(48 123)(49 97)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 140)(66 139)(67 138)(68 137)(69 136)(70 135)(71 134)(72 133)(73 132)(74 131)(75 130)(76 129)(77 144)(78 143)(79 142)(80 141)

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

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

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

39 conjugacy classes

class 1 2A2B2C3A3B3C3D 4 6A6B6C6D6E···6L8A8B12A12B12C12D16A16B16C16D24A···24H
order12223333466666···688121212121616161624···24
size118722222222228···8224444181818184···4

39 irreducible representations

dim111122222244
type+++++++++++
imageC1C2C2C2S3D4D6D8C3⋊D4D16D4⋊S3C3⋊D16
kernelC327D16C24.S3C325D8C32×D8C3×D8C3×C12C24C3×C6C12C32C6C3
# reps111141428448

Matrix representation of C327D16 in GL6(𝔽97)

0960000
1960000
001000
000100
000010
000001
,
9610000
9600000
0096100
0096000
000010
000001
,
15410000
56820000
0009600
0096000
00002452
00007169
,
010000
100000
000100
001000
00009695
000001

G:=sub<GL(6,GF(97))| [0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[96,96,0,0,0,0,1,0,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,56,0,0,0,0,41,82,0,0,0,0,0,0,0,96,0,0,0,0,96,0,0,0,0,0,0,0,24,71,0,0,0,0,52,69],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,96,0,0,0,0,0,95,1] >;

C327D16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_7D_{16}
% in TeX

G:=Group("C3^2:7D16");
// GroupNames label

G:=SmallGroup(288,301);
// by ID

G=gap.SmallGroup(288,301);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,254,135,142,675,346,80,2693,9414]);
// Polycyclic

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

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