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G = C9⋊D16order 288 = 25·32

The semidirect product of C9 and D16 acting via D16/D8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C92D16, D81D9, D723C2, C8.4D18, C24.6D6, C36.3D4, C18.8D8, C72.2C22, C9⋊C161C2, (C9×D8)⋊1C2, C3.(C3⋊D16), (C3×D8).1S3, C2.4(D4⋊D9), C4.1(C9⋊D4), C6.15(D4⋊S3), C12.1(C3⋊D4), SmallGroup(288,33)

Series: Derived Chief Lower central Upper central

C1C72 — C9⋊D16
C1C3C9C18C36C72D72 — C9⋊D16
C9C18C36C72 — C9⋊D16
C1C2C4C8D8

Generators and relations for C9⋊D16
 G = < a,b,c | a9=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >

8C2
72C2
4C22
36C22
8C6
24S3
2D4
18D4
4C2×C6
12D6
8C18
8D9
9D8
9C16
2C3×D4
6D12
4C2×C18
4D18
9D16
3D24
3C3⋊C16
2D36
2D4×C9
3C3⋊D16

Smallest permutation representation of C9⋊D16
On 144 points
Generators in S144
(1 70 55 86 27 119 136 47 108)(2 109 48 137 120 28 87 56 71)(3 72 57 88 29 121 138 33 110)(4 111 34 139 122 30 89 58 73)(5 74 59 90 31 123 140 35 112)(6 97 36 141 124 32 91 60 75)(7 76 61 92 17 125 142 37 98)(8 99 38 143 126 18 93 62 77)(9 78 63 94 19 127 144 39 100)(10 101 40 129 128 20 95 64 79)(11 80 49 96 21 113 130 41 102)(12 103 42 131 114 22 81 50 65)(13 66 51 82 23 115 132 43 104)(14 105 44 133 116 24 83 52 67)(15 68 53 84 25 117 134 45 106)(16 107 46 135 118 26 85 54 69)
(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)(17 113)(18 128)(19 127)(20 126)(21 125)(22 124)(23 123)(24 122)(25 121)(26 120)(27 119)(28 118)(29 117)(30 116)(31 115)(32 114)(33 53)(34 52)(35 51)(36 50)(37 49)(38 64)(39 63)(40 62)(41 61)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(65 97)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 141)(82 140)(83 139)(84 138)(85 137)(86 136)(87 135)(88 134)(89 133)(90 132)(91 131)(92 130)(93 129)(94 144)(95 143)(96 142)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C8A8B9A9B9C 12 16A16B16C16D18A18B18C18D···18I24A24B36A36B36C72A···72F
order12223466688999121616161618181818···18242436363672···72
size1187222288222224181818182228···8444444···4

39 irreducible representations

dim11112222222224444
type+++++++++++++++
imageC1C2C2C2S3D4D6D8D9C3⋊D4D16D18C9⋊D4D4⋊S3C3⋊D16D4⋊D9C9⋊D16
kernelC9⋊D16C9⋊C16D72C9×D8C3×D8C36C24C18D8C12C9C8C4C6C3C2C1
# reps11111112324361236

Matrix representation of C9⋊D16 in GL4(𝔽433) generated by

3503600
39738600
0010
0001
,
0100
1000
0021482
00392132
,
0100
1000
0010
00432432
G:=sub<GL(4,GF(433))| [350,397,0,0,36,386,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,214,392,0,0,82,132],[0,1,0,0,1,0,0,0,0,0,1,432,0,0,0,432] >;

C9⋊D16 in GAP, Magma, Sage, TeX

C_9\rtimes D_{16}
% in TeX

G:=Group("C9:D16");
// GroupNames label

G:=SmallGroup(288,33);
// by ID

G=gap.SmallGroup(288,33);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,254,135,142,675,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^9=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C9⋊D16 in TeX

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